Overview of protein structure

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- [Voiceover] We've already spent a lot of time talking

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about proteins, and how they do a huge variety of things

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in biological systems,

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anything from acting as hormones to antibodies

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to providing structures in cells, signaling mechanism,

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a whole series of things and their ability to do

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all of those things in living systems

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comes out, it's a by-product of their structure,

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so what we want to talk about in this video

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is protein,

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protein structure, and to just get a high-level appreciation

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for protein structure, this is a hemoglobin molecule

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right over here,

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and this hemoglobin molecule,

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it's made out of four polypeptide chains.

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Two of them have 141 amino acids,

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two of them have 146 amino acids,

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for a total of 574,

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574 amino acids.

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But you see, they don't just go into some random

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configuration, they come into a configuration that is really

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good for doing what hemoglobin does,

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and that is being a transporter for oxygen,

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being a transporter for oxygen within red blood cells.

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So how do proteins like hemoglobin,

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there's many, many other types of proteins

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that do many, many other types of things.

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How do they get their structure?

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Well, one way to think about it is,

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there's different layers of the structure,

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or there's different degrees of structure.

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The first degree of the structure we can call

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the primary structure.

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Primary structure, and this is really just the sequence

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of the amino acids.

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When we talk about the translation step,

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when we go from mRNA and we go to a ribosome

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and the tRNA brings the amino acids and puts them,

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and starts linking them together,

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it's setting up the primary,

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it's setting up the primary structure.

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The DNA, the information in DNA,

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that's essentially what it's coding.

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It's coding for what order do we put the different amino

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acids in, so this is just the order,

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the order,

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order of,

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of the amino,

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of the amino acids.

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Now the next level of structure,

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this is just the order, this is how we form our polypeptide,

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but how does a polypeptide start getting bent into

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these shapes to be able to do the different things

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that it needs to do?

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Well the second,

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the second order of our structure,

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or I could say the secondary structure,

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secondary structure,

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this is due to interactions of the peptide backbone.

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Due to interactions,

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interactions of the backbone

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of the peptide,

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of the peptide backbone,

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and I have some examples of that.

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I have some examples of that right over here.

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You see, right over here, we have a bunch of,

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we have a polypeptide chain.

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We have a bunch of amino acids that are bonded

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with the peptide bonds.

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This is a,

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this is a peptide bond over here.

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This is a,

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this is between the carbonyl carbon and the nitrogen,

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another peptide bond between the carbonyl carbon

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and the nitrogen, another peptide bond.

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And this chain,

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this polypeptide chain, you can imagine maybe it goes

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down here,

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maybe it goes around, maybe it comes back, who knows?

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But we see that when it comes back,

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we still are going from nitrogen

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to carbonyl carbon, polypeptide linkage,

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nitrogen, carbonyl carbon,

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peptide linkage,

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but what you see happening is,

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from these two chains, the backbones are interacting.

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I actually didn't even explicitly even draw the side chains.

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I just put an "R" here for the different side chains,

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but you see how they're interacting.

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Right over here, we have nitrogen.

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Nitrogen is electronegative.

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It would hog the electrons from the hydrogen,

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so the hydrogen's going to have a partially positive charge.

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Oxygen is electronegative.

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It's going to hog the electrons from the carbon,

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so it's going to have a partially negative charge,

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and so this hydrogen, this oxygen,

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they're going to be attracted to each other.

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This is a hydrogen bond,

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our good, old friend the hydrogen bond.

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Same thing is going to happen over here,

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same thing is going to happen over here.

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And so these two chains,

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these can form kind of this sheet,

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in fact it's called,

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this is called a beta-pleated sheet.

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Beta-pleated,

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beta-pleated sheet.

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Now, over here, I have also constructred a beta-pleated

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sheet but you might notice the difference.

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This one went from the nitrogen to the alpha carbon,

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this is the alpha carbon over here,

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to the carbonyl carbon, nitrogen, alpha carbon,

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carbonyl carbon, and this one was also going in the same

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direction, nitrogen,

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alpha carbon, carbonyl carbon, nitrogen,

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alpha carbon, carbonyl carbon,

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so this is a parallel beta-pleated sheet.

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These, both of these side chains that are interacting,

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sorry, both of these backbones that are interacting

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are going in the same direction,

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so we would just call this one,

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we would just call this one a parallel beta-pleated sheet.

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Now this one, they're parallel, but what we see going on,

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we go nitrogen, alpha carbon, carbonyl carbon,

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nitrogen, alpha carbon, carbonyl carbon,

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that's on the left side,

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but on the right side, we're going carbonyl carbon,

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alpha carbon, nitrogen.

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Carbonyl carbon, alpha carbon, nitrogen.

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We're going in the opposite direction.

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In fact, even to construct this,

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I copy and pasted this but I rotated it around.

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You can see that I've actually drawn it upside down,

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and so here, we have these two things,

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you still have the hydrogen bonds between these,

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between these partially positive ends of these,

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of this bond, this nitrogen-hydrogen bond,

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at the hydrogen end,

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and this partially negative charge of the oxygen.

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You still see,

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you still have these hydrogen bonds,

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but, and these backbones are parallel,

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but they are going in different directions.

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They are, so we would say these are anti,

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these are anti-parallel beta-pleated sheets.

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So, anti,

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anti-parallel beta-pleated sheets.

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So this is another form of a secondary structure.

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Now this over here, we see,

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we see that the backbone is going in this,

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it's going in this, in this helical structure,

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and we have, essentially, hydrogen bonds between

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the different layers of the helices,

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or between the different layers of the helix, I should say.

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So, over here, this oxygen is partially negative,

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partially negative charge.

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This hydrogen, partially positive charge,

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so I have a hydrogen bond.

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I could have a hydrogen bond over here,

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and so that's what gives this a helical,

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a helical structure,

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and we would call this an alpha,

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an alpha helix,

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so these interactions between the backbone,

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between the backbone, the peptide backbone,

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that's the secondary structure.

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That's the secondary structure of a protein.

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Now, we're not done yet because you could imagine

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these side chains have something to say.

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Some of these side chains are hydrophobic,

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so they would want to kind of pull that part into, kind of,

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away from, if it's in water, away from the outside.

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Some of these side chains might form hydrogen bonds

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with other side chains, so they,

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those would interact in certain ways.

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You could have side chains that form, actually,

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disulfide bonds,

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actually covalent bonds with other side chains,

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and we're gonna go into a lot more detail of that

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in a future video but the,

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the third, I guess, the third form of structure,

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and we'll call that the tertiary structure,

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the tertiary structure,

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this is due to interactions of side chains,

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so, due to side chain,

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due to side chain interactions.

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Due to side chain interactions.

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And so you can imagine, you know, if, let's say I have

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this, you know, this thing over here, maybe,

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maybe I have a bunch of hydrophobic,

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hydrophobic side chains right over here.

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I'm just drawing them as "R"

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but let's just assume they're hydrophobic

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and let's say that the H2O is out here,

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they might, and we'd have to think in three dimensions,

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but they might want to get away from the H2O,

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and likewise, you can maybe have hydrophilic side chains,

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maybe polar side chains might be on,

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might be on the outside.

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You might have a situation where,

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where one side chain,

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let's call it R1 and another side chain R2,

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maybe they form hydrogen bonds with each other.

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Maybe they have an ionic bond with each other,

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so there's a bunch of different types of side chain

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interactions that we could actually think about,

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and we'll go into more depth in that in a future video.

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Now, any protein that's made up of a single polypeptide

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is only going to have primary structure, secondary structure

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and tertiary structure,

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but if we're dealing with something like hemoglobin,

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that's made up of more than one polypeptide,

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then we're going to talk about quaternary structure.

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So, quaternary,

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quaternary, quaternary structure,

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quaternary structure,

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and this is all about how the different polypeptide chains

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come together to form the larger,

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to form the larger complex,

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so, multiple,

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multiple, I guess you could say, interactions or arrangement

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of multiple chains together.

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So, arrangement,

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arrangement of multiple,

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multiple, in the case of hemoglobin we had four,

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multiple peptide, peptide chains.

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So hopefully that gives you an appreciation,

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and this is a fascinating thing.

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Protein structure is a fascinating area,

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in fact, there are so many permutations,

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so how you can, how you can actually construct proteins,

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that if we understand that better, we'll be able to,

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much better be able to go from DNA,

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to be able to translate to primary structure,

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and then to really figure out how proteins work,

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what they do, how they can be fixed,

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how they can maybe provide other functions,

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so it's a fascinating, fascinating field of study.


Introduction to amino acids

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Kind: captions
Language: en

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- DNA gets a lot of attention as the store

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of our genetic information, and it deserves that.

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If we didn't have DNA, there would be no way

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of keeping the information that makes us us,

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and other organisms what those organisms are.

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And DNA has some neat properties, it can replicate itself,

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and we go into a lot of depth on that in other videos.

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So DNA producing more DNA, we call that,

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we call that replication, but just being able

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to replicate yourself on its own isn't enough

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to actually produce an organism.

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And to produce an organism, you somehow have to

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take that information in the DNA, and then

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produce things like a structural molecules,

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enzymes, transport molecules, signaling molecules,

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that actually do the work of the organism.

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And that process, the first step, and this is all a review

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that we've seen in other videos.

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The first step is to go from DNA to RNA,

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and in particular, messenger RNA.

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"Messenger RNA," and this process right over here,

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this is called transcription.

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"Transcription," we go into a lot of detail

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on this in other videos.

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And then you wanna go from that messenger RNA,

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it goes to the ribosomes and then tRNA goes and grabs

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amino acids, and they form actual proteins.

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So you go from messenger RNA, and then in conjunction,

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so this is all, this is in conjunction with tRNA

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and amino acids, so let me say "+tRNA," and "amino acids."

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And I'll write "amino acids" in, I'll write it in a brighter

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color, since that's going to be the focus of this video.

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So tRNA and amino acids, you're able to construct proteins.

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You are able to construct proteins,

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which are made up of chains of amino acids,

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and it's the proteins that do

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a lot of the work of the organism.

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Proteins, which are nothing but chains of amino acids,

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or they're made up of, sometimes

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multiple chains of amino acids.

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So you can image, I'm just going to, that's an amino acid.

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That's another amino acid.

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This is an amino acid.

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This is an amino acid, you could keep going.

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So these chains of amino acids, based on

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how these different, based on the properties

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of these different amino acids,

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and how the protein takes shape and how

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it might interact with its surrounding,

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these proteins can serve all sorts of different functions.

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Anything from part of your immune system,

00:02:30.619 --> 00:02:33.324
antibodies, they can serve as enzymes,

00:02:33.324 --> 00:02:38.420
they can serve as signaling hormones, like insulin.

00:02:38.420 --> 00:02:40.986
They're involved in muscle contraction.

00:02:40.986 --> 00:02:42.287
Actin and myosin, we actually have

00:02:42.287 --> 00:02:43.726
a fascinating video on that.

00:02:43.726 --> 00:02:45.491
Transport of oxygen.

00:02:45.491 --> 00:02:46.455
Hemoglobin.

00:02:46.455 --> 00:02:48.719
So proteins, the way at least my brain of it,

00:02:48.719 --> 00:02:50.089
is they do a lot of the work.

00:02:50.089 --> 00:02:54.419
DNA says, well, what contains the information,

00:02:54.419 --> 00:02:57.809
but a lot of the work of organism is actually done,

00:02:57.809 --> 00:03:00.526
is actually done by the proteins.

00:03:00.526 --> 00:03:02.349
And as I just said, the building blocks

00:03:02.349 --> 00:03:05.054
of the proteins are the amino acids.

00:03:05.054 --> 00:03:07.027
So let's focus on that a little bit.

00:03:07.027 --> 00:03:10.859
So up here are some examples of amino acids.

00:03:10.859 --> 00:03:12.786
And there are 20 common amino acids,

00:03:12.786 --> 00:03:17.523
there are a few more depending on what organism you look at,

00:03:17.523 --> 00:03:19.740
and theoretically there could be many more.

00:03:19.740 --> 00:03:21.087
But in most biological systems,

00:03:21.087 --> 00:03:24.826
there are 20 common amino acids that the DNA is coding for,

00:03:24.826 --> 00:03:26.219
and these are two of them.

00:03:26.219 --> 00:03:29.017
So let's just first look at what is common.

00:03:29.017 --> 00:03:33.022
So, we see that both these, and actually all three of this,

00:03:33.022 --> 00:03:36.761
this is just a general form, you have an amino group.

00:03:36.761 --> 00:03:39.059
You have an amino group, and this where,

00:03:39.059 --> 00:03:42.461
this is why we call it an "amino," an amino acid.

00:03:42.461 --> 00:03:44.423
So you have an amino group.

00:03:44.423 --> 00:03:46.258
Amino group right over here.

00:03:46.258 --> 00:03:49.056
Now you might say, "well, it's called an amino acid,"

00:03:49.056 --> 00:03:50.960
"so where is the acid?"

00:03:50.960 --> 00:03:55.720
And that comes from this carboxyl group right over here.

00:03:55.720 --> 00:03:57.392
So that's why we call it an acid.

00:03:57.392 --> 00:04:02.384
This carboxyl group is acidic.

00:04:02.384 --> 00:04:04.520
It likes to donate this proton.

00:04:04.520 --> 00:04:06.656
And then in between, we have a carbon,

00:04:06.656 --> 00:04:09.152
and we call that the alpha carbon.

00:04:09.152 --> 00:04:12.020
We call that the alpha carbon.

00:04:12.020 --> 00:04:14.493
Alpha carbon, and that alpha carbon is bonded,

00:04:14.493 --> 00:04:16.594
it has a covalent bond to the amino group,

00:04:16.594 --> 00:04:18.591
covalent bond to the carboxyl group,

00:04:18.591 --> 00:04:20.960
and a covalent bond to a hydrogen.

00:04:20.960 --> 00:04:24.187
Now, from there, that's where you get the variation

00:04:24.187 --> 00:04:26.242
in the different amino acids, and actually,

00:04:26.242 --> 00:04:30.294
there's even some exceptions for how the nitrogen is,

00:04:30.294 --> 00:04:32.825
but for the most part, the variation between

00:04:32.825 --> 00:04:36.018
the amino acids is what this fourth covalent bond

00:04:36.018 --> 00:04:38.189
from the alpha carbon does.

00:04:38.189 --> 00:04:43.321
So you see in serine, you have this,

00:04:43.321 --> 00:04:45.155
what you could call it an alcohol.

00:04:45.155 --> 00:04:47.187
You could have an alcohol side chain.

00:04:47.187 --> 00:04:50.728
In valine right over here, you have a

00:04:50.728 --> 00:04:54.917
fairly pure hydrocarbon, hydrocarbon side chain.

00:04:54.917 --> 00:04:58.158
And so in general, we refer to these side chains

00:04:58.158 --> 00:05:01.154
as an R group, and it's these R groups

00:05:01.154 --> 00:05:05.159
that play a big role in defining the shape of the proteins,

00:05:05.159 --> 00:05:07.051
and how they interact with their environment

00:05:07.051 --> 00:05:08.955
and the types of things they can do.

00:05:08.955 --> 00:05:10.755
And you can even see, just from these examples

00:05:10.755 --> 00:05:13.518
how these different sides chains might behave differently.

00:05:13.518 --> 00:05:16.084
This one has an alcohol side chain,

00:05:16.084 --> 00:05:18.859
and we know that oxygen is electronegative,

00:05:18.859 --> 00:05:21.378
it likes to hog electrons, it's amazing how much

00:05:21.378 --> 00:05:23.979
of chemistry or even biology you can deduce

00:05:23.979 --> 00:05:26.220
from just pure electronegativity.

00:05:26.220 --> 00:05:28.750
So, oxygen likes to hog electrons, so you're gonna have

00:05:28.750 --> 00:05:31.757
a partially-negative charge there.

00:05:31.757 --> 00:05:38.259
Hydrogen has a low electronegativity relative to oxygen,

00:05:38.259 --> 00:05:40.291
so it's gonna have its electrons hogged,

00:05:40.291 --> 00:05:42.822
so you're gonna have a partially positive charge,

00:05:42.822 --> 00:05:46.781
just like that, and so this has a polarity to it,

00:05:46.781 --> 00:05:48.789
and so it's going to be hydrophilic, it's going to,

00:05:48.789 --> 00:05:50.693
at least this part of the molecule is going to

00:05:50.693 --> 00:05:54.153
be able to be attracted and interact with water.

00:05:54.153 --> 00:05:57.288
And that's in comparison to what we have over here,

00:05:57.288 --> 00:06:02.489
this hydrocarbon side chain, this has no polarity over here,

00:06:02.489 --> 00:06:04.683
so this is going to be hydrophobic.

00:06:04.683 --> 00:06:08.015
So this is going to be hydrophobic.

00:06:08.015 --> 00:06:10.523
And so when we start talking about the structures

00:06:10.523 --> 00:06:12.625
of proteins, and how the structures of proteins

00:06:12.625 --> 00:06:14.552
are influenced by its side chains,

00:06:14.552 --> 00:06:17.524
you could image that parts of proteins that have

00:06:17.524 --> 00:06:18.917
hydrophobic side chains, those are gonna

00:06:18.917 --> 00:06:21.460
wanna get onto the inside of the proteins

00:06:21.460 --> 00:06:23.561
if we're in an aqueous solution,

00:06:23.561 --> 00:06:24.722
while the ones that are more hydrophilic

00:06:24.722 --> 00:06:26.255
will wanna go onto the outside,

00:06:26.255 --> 00:06:27.427
and you might have some side chains

00:06:27.427 --> 00:06:29.250
that are all big and bulky, and so they might

00:06:29.250 --> 00:06:32.420
make it hard to tightly pack, and then you might have

00:06:32.420 --> 00:06:34.080
other side chains that are nice and small

00:06:34.080 --> 00:06:36.355
that make it very easy to pack, so these things

00:06:36.355 --> 00:06:39.653
really do help define the shape,

00:06:39.653 --> 00:06:41.487
and we're gonna talk about that a lot more

00:06:41.487 --> 00:06:44.482
when we talk about the structure.

00:06:44.482 --> 00:06:47.861
But how do these things actually connect?

00:06:47.861 --> 00:06:49.324
And we're gonna go into much more detail

00:06:49.324 --> 00:06:51.924
in another video, but if you have...

00:06:51.924 --> 00:06:56.684
If you have serine right over here, and then you have

00:06:56.684 --> 00:07:01.514
valine right over here, they connect through

00:07:01.514 --> 00:07:03.558
what we call peptide bonds, and a peptide

00:07:03.558 --> 00:07:08.515
is the term for two or more amino acids connected together,

00:07:08.515 --> 00:07:10.257
so this would be a dipeptide, and the bond

00:07:10.257 --> 00:07:12.253
isn't this big, I just, actually let me just,

00:07:12.253 --> 00:07:14.692
let me draw it a little bit smaller.

00:07:14.692 --> 00:07:15.585
So...

00:07:15.585 --> 00:07:17.455
That's serine.

00:07:17.455 --> 00:07:20.624
This is valine.

00:07:20.624 --> 00:07:23.654
They can form a peptide bond, and this would be the smallest

00:07:23.654 --> 00:07:27.428
peptide, this would be a dipeptide right over here.

00:07:27.428 --> 00:07:30.052
"Peptide," "peptide bond," or sometimes

00:07:30.052 --> 00:07:32.141
called a peptide linkage.

00:07:32.141 --> 00:07:35.253
And as this chain forms, that polypeptide,

00:07:35.253 --> 00:07:37.610
as you add more and more things to it,

00:07:37.610 --> 00:07:42.660
as you add more and more amino acids,

00:07:42.660 --> 00:07:44.518
this is going to be, this can be a protein

00:07:44.518 --> 00:07:47.687
or can be part of a protein that does all of these things.

00:07:47.687 --> 00:07:50.021
Now one last thing I wanna talk about,

00:07:50.021 --> 00:07:52.888
this is the way, the way these amino acids have been drawn

00:07:52.888 --> 00:07:54.920
is a way you'll often see them in a textbook,

00:07:54.920 --> 00:07:59.193
but at physiological pH's, the pH's inside of your body,

00:07:59.193 --> 00:08:01.956
which is in that, you know, that low sevens range,

00:08:01.956 --> 00:08:08.318
so it's a pH of roughly 7.2 to 7.4.

00:08:08.318 --> 00:08:12.985
What you have is this, the carboxyl group right over here,

00:08:12.985 --> 00:08:15.087
is likely to be deprotonated, it's likely

00:08:15.087 --> 00:08:16.991
to have given away its hydrogen,

00:08:16.991 --> 00:08:19.452
you're gonna find that more likely than when you have...

00:08:19.452 --> 00:08:21.019
It's gonna be higher concentrations having been

00:08:21.019 --> 00:08:23.121
deprotonated than being protonated.

00:08:23.121 --> 00:08:26.894
So, at physiological conditions, it's more likely

00:08:26.894 --> 00:08:33.546
that this oxygen has taken both of those electrons,

00:08:33.546 --> 00:08:35.125
and now has a negative charge,

00:08:35.125 --> 00:08:37.947
so it's given, it's just given away the hydrogen proton

00:08:37.947 --> 00:08:40.153
but took that hydrogen's electron.

00:08:40.153 --> 00:08:43.717
So it might be like this, and then the amino group,

00:08:43.717 --> 00:08:46.689
the amino group at physiological pH's,

00:08:46.689 --> 00:08:49.150
it's likely to actually grab a proton.

00:08:49.150 --> 00:08:52.691
So nitrogen has an extra loan pair,

00:08:52.691 --> 00:08:55.153
so it might use that loan pair to grab a proton,

00:08:55.153 --> 00:08:56.952
in fact it's physiological pH's,

00:08:56.952 --> 00:08:58.752
you'll find a higher concentration of it having

00:08:58.752 --> 00:09:01.457
grabbed a proton than not grabbing a proton.

00:09:01.457 --> 00:09:05.695
So, the nitrogen will have grabbed a proton,

00:09:05.695 --> 00:09:07.587
use its loan pairs to grab a proton,

00:09:07.587 --> 00:09:09.793
and so it is going to have...

00:09:09.793 --> 00:09:11.825
So it is going to have a...

00:09:11.825 --> 00:09:14.321
It is going to have a positive charge.

00:09:14.321 --> 00:09:19.394
And so sometimes you will see amino acids

00:09:19.394 --> 00:09:21.217
described this way, and this is actually more accurate

00:09:21.217 --> 00:09:24.491
for what you're likely to find at physiological conditions,

00:09:24.491 --> 00:09:26.256
and these molecules have an interesting name,

00:09:26.256 --> 00:09:28.752
a molecule that is neutral even though

00:09:28.752 --> 00:09:31.120
parts of it have charge, like this,

00:09:31.120 --> 00:09:33.419
this is called a zwitterion.

00:09:33.419 --> 00:09:35.985
That's a fun, fun word.

00:09:35.985 --> 00:09:39.874
Zwitterion.

00:09:39.874 --> 00:09:41.558
And "zwitter" in German means "hybrid,"

00:09:41.558 --> 00:09:44.809
and "ion" obviously means that it's going to have charge,

00:09:44.809 --> 00:09:46.086
and so this has hybrid charge,

00:09:46.086 --> 00:09:47.990
even though it has charges at these ends,

00:09:47.990 --> 00:09:51.293
the charges net out to be neutral.


Peptide bond formation

WEBVTT
Kind: captions
Language: en

00:00:00.483 --> 00:00:03.191
- [Voiceover] So I've got two arbitrary amino acids here.

00:00:03.191 --> 00:00:05.987
We recognize the telltale signs of an amino acid.

00:00:05.987 --> 00:00:08.727
We have an amino group right over here

00:00:08.727 --> 00:00:10.980
that gives us the amino and amino acid.

00:00:10.980 --> 00:00:13.393
We have a carboxyl group right over here.

00:00:13.393 --> 00:00:15.831
This is the acid part of an amino acid.

00:00:15.831 --> 00:00:17.527
And in between we have a carbon,

00:00:17.527 --> 00:00:21.084
and we call that the alpha carbon.

00:00:21.084 --> 00:00:22.983
And that alpha carbon is gonna be bonded

00:00:22.983 --> 00:00:26.444
to a hydrogen and some type of a side chain,

00:00:26.444 --> 00:00:28.882
and we're just gonna call this side chain R1,

00:00:28.882 --> 00:00:31.342
and then we're gonna call this side chain R2.

00:00:31.342 --> 00:00:33.177
And what we're gonna concern ourselves with

00:00:33.177 --> 00:00:36.172
in this video is, how do you take two amino acids

00:00:36.172 --> 00:00:38.286
and form a peptide out of them?

00:00:38.286 --> 00:00:41.157
And just as a reminder, a peptide is nothing more...

00:00:43.927 --> 00:00:46.150
than a chain of amino acids.

00:00:55.750 --> 00:00:59.786
And so, how do you take these two amino acids

00:00:59.786 --> 00:01:01.992
and form a dipeptide like this?

00:01:01.992 --> 00:01:03.920
A dipeptide would have two amino acids.

00:01:03.920 --> 00:01:05.963
That would be the smallest possible peptide,

00:01:05.963 --> 00:01:07.797
but then you could keep adding amino acids

00:01:07.797 --> 00:01:09.631
and form polypeptides.

00:01:09.631 --> 00:01:12.761
And a very high-level overview of this reaction is

00:01:12.761 --> 00:01:15.181
that this nitrogen uses its lone pair

00:01:15.181 --> 00:01:20.012
to form a bond with this carbonyl carbon right over here.

00:01:20.382 --> 00:01:24.143
So this lone pair goes to this carbonyl carbon,

00:01:24.143 --> 00:01:27.278
forms a bond, and then this hydrogen,

00:01:27.278 --> 00:01:31.432
this hydrogen, and this oxygen could be used net net

00:01:31.432 --> 00:01:33.738
to form a water molecule...

00:01:35.208 --> 00:01:39.235
that's let go from both of these amino acids.

00:01:39.235 --> 00:01:41.161
So this reaction, you end up

00:01:41.161 --> 00:01:44.319
with the nitrogen being attached to this carbon,

00:01:44.319 --> 00:01:47.732
and a release of a water molecule.

00:01:47.732 --> 00:01:50.356
And because you have the release of this water molecule,

00:01:50.356 --> 00:01:52.818
this type of reaction, and we've seen it many other times

00:01:52.818 --> 00:01:55.698
with other types of molecules,

00:01:55.698 --> 00:01:57.995
we call this a condensation reaction,

00:01:57.995 --> 00:02:00.178
or a dehydration synthesis.

00:02:00.178 --> 00:02:03.314
So condensation...

00:02:04.734 --> 00:02:06.764
condensation reaction

00:02:07.264 --> 00:02:10.135
or dehydration synthesis.

00:02:11.515 --> 00:02:12.820
We saw this type of reaction

00:02:12.820 --> 00:02:14.681
when we were putting glucoses together,

00:02:14.681 --> 00:02:17.351
when we were forming carbohydrates.

00:02:17.351 --> 00:02:21.484
Dehydration synthesis.

00:02:21.484 --> 00:02:23.133
But whenever I see a reaction like this,

00:02:23.133 --> 00:02:24.874
it's somewhat satisfying to just be able

00:02:24.874 --> 00:02:27.057
to do the counting and say, "All right, this is gonna bond

00:02:27.057 --> 00:02:29.170
"with that, we see the bond right over there,

00:02:29.170 --> 00:02:32.305
"and I'm gonna let go of an oxygen and two hydrogens,

00:02:32.305 --> 00:02:35.950
"which net net equals H2O, equals a water molecule."

00:02:35.950 --> 00:02:37.854
But how can we actually imagine this happening?

00:02:37.854 --> 00:02:39.178
Can we push the electrons around?

00:02:39.178 --> 00:02:42.306
Can we do a little bit of high-level organic chemistry

00:02:42.966 --> 00:02:44.309
to think about how this happens?

00:02:44.309 --> 00:02:45.374
And that's what I wanna do here.

00:02:45.374 --> 00:02:47.560
I'm not gonna do a formal reaction mechanism,

00:02:47.560 --> 00:02:49.928
but really get a sense of what's going on.

00:02:49.928 --> 00:02:51.994
Well, nitrogen, as we said, has got this lone pair,

00:02:51.994 --> 00:02:53.433
it's electronegative.

00:02:53.433 --> 00:02:55.663
And this carbon right over here, it's attached

00:02:55.663 --> 00:02:58.450
to two oxygens, oxygens are more electronegative.

00:02:58.450 --> 00:03:01.049
The oxygens might hog those electrons.

00:03:01.049 --> 00:03:03.210
And so this nitrogen might wanna do

00:03:03.210 --> 00:03:05.950
what we call in organic chemistry a nucleophilic attack

00:03:05.950 --> 00:03:08.294
on this carbon right over here.

00:03:08.294 --> 00:03:09.849
And when it does that,

00:03:09.849 --> 00:03:12.102
if we were doing a more formal reaction mechanism,

00:03:12.102 --> 00:03:13.657
we could say, "Hey, well, maybe one

00:03:13.657 --> 00:03:16.978
"of the double bonds goes back,

00:03:16.978 --> 00:03:19.346
"the electrons in it go back to this oxygen,

00:03:19.346 --> 00:03:23.177
"and then that oxygen would have a negative charge."

00:03:23.177 --> 00:03:24.316
But then that lone pair

00:03:24.316 --> 00:03:27.484
from that double bond could then reform,

00:03:27.484 --> 00:03:30.505
and as that happens,

00:03:30.505 --> 00:03:35.415
this oxygen that's in the hydroxyl group will take back both

00:03:35.415 --> 00:03:37.087
of these electrons.

00:03:37.087 --> 00:03:40.940
Would take back both of those electrons,

00:03:40.940 --> 00:03:45.088
and now it's going to have an extra lone pair.

00:03:45.088 --> 00:03:47.884
Let me do that by erasing this bond

00:03:47.884 --> 00:03:50.972
and then giving it an extra lone pair.

00:03:50.972 --> 00:03:54.344
It already had two lone pairs,

00:03:54.344 --> 00:03:55.430
and then when it took that bond,

00:03:55.430 --> 00:03:57.467
it's gonna have a third lone pair.

00:03:57.467 --> 00:04:01.234
And then it's going to have a negative charge.

00:04:01.234 --> 00:04:02.489
And now you could imagine it's going

00:04:02.489 --> 00:04:04.554
to grab a hydrogen proton someplace.

00:04:04.554 --> 00:04:06.274
And now it could just grab any hydrogen proton,

00:04:06.274 --> 00:04:09.082
but probably the most convenient one would be this one,

00:04:09.082 --> 00:04:12.472
because if this nitrogen is going to use this lone pair

00:04:12.472 --> 00:04:14.006
to form a bond with carbon,

00:04:14.006 --> 00:04:16.303
it's going to have a positive charge,

00:04:16.303 --> 00:04:18.951
and it might wanna take these electrons back.

00:04:18.951 --> 00:04:21.041
So you could imagine where one

00:04:21.041 --> 00:04:23.966
of these lone pairs is used to grab this hydrogen proton,

00:04:23.966 --> 00:04:27.796
and then the nitrogen can take these electrons,

00:04:27.796 --> 00:04:29.576
can take these electrons back.

00:04:29.576 --> 00:04:31.652
So hopefully you didn't find this too convoluted,

00:04:31.652 --> 00:04:32.581
but I always like to think,

00:04:32.581 --> 00:04:33.881
what could actually happen here?

00:04:33.881 --> 00:04:37.038
And so you see, this lone pair of electrons

00:04:37.038 --> 00:04:39.849
from the nitrogen forms this orange bond with the carbon.

00:04:43.239 --> 00:04:44.421
Let me do that in orange color

00:04:44.421 --> 00:04:46.140
if I'm going to call it an orange bond.

00:04:46.140 --> 00:04:47.950
It forms this orange bond.

00:04:48.970 --> 00:04:50.409
What we call this orange bond,

00:04:50.409 --> 00:04:54.821
we could call this a peptide bond, or a peptide linkage.

00:04:54.821 --> 00:04:59.418
Peptide bond,

00:04:59.418 --> 00:05:01.228
sometimes called a peptide...

00:05:03.508 --> 00:05:05.378
peptide linkage.

00:05:05.388 --> 00:05:10.238
And then we have the release of a water molecule.

00:05:10.238 --> 00:05:12.676
And so you have this oxygen is this oxygen,

00:05:12.676 --> 00:05:16.089
and you could imagine that this hydrogen is this hydrogen,

00:05:16.089 --> 00:05:19.897
and this hydrogen is this hydrogen right over here,

00:05:19.897 --> 00:05:21.802
and so net net it all works out.

00:05:21.802 --> 00:05:23.380
Now when I first saw this reaction,

00:05:23.380 --> 00:05:25.447
I was like, "OK, that kind of makes sense."

00:05:25.447 --> 00:05:28.906
Except for the fact that in physiological pHs,

00:05:28.906 --> 00:05:32.203
amino acids don't tend to be in this form.

00:05:32.203 --> 00:05:35.792
In physiological pHs, you are more likely

00:05:35.792 --> 00:05:38.635
to find this form of the amino acids,

00:05:38.635 --> 00:05:42.489
to find them as zwitterions.

00:05:42.489 --> 00:05:44.557
Zwitterions. Let me write down that word.

00:05:44.557 --> 00:05:45.829
It's a fun word to say.

00:05:47.139 --> 00:05:48.828
And it's one word, but I'm gonna write the two parts

00:05:48.828 --> 00:05:50.616
of the word in different colors so you can see.

00:05:50.616 --> 00:05:52.404
It's a zwitterion.

00:05:52.404 --> 00:05:53.425
So what does that mean?

00:05:53.425 --> 00:05:55.702
Well, zwitter in German means hybrid.

00:05:55.702 --> 00:05:56.769
It's a hybrid ion.

00:05:56.769 --> 00:06:00.647
It's an ion, it has charge on different ends of it,

00:06:00.647 --> 00:06:03.781
parts of the molecule have charge,

00:06:03.781 --> 00:06:06.776
but when you net 'em out, it is neutral.

00:06:07.386 --> 00:06:10.446
Parts are charged, but is neutral overall.

00:06:10.446 --> 00:06:14.974
And so at physiological pHs, the amino,

00:06:14.974 --> 00:06:17.481
the nitrogen end of the amino acid,

00:06:17.481 --> 00:06:20.940
tends to grab an extra proton, becomes a positive charge,

00:06:20.940 --> 00:06:25.306
and the carboxyl group tends to let go of a proton

00:06:25.306 --> 00:06:27.048
and has a negative charge.

00:06:27.048 --> 00:06:29.462
And this is actually going to be in equilibrium

00:06:29.462 --> 00:06:31.343
with the forms that we just saw before,

00:06:31.343 --> 00:06:33.781
but at physiological pHs, it will actually tend

00:06:33.781 --> 00:06:37.240
to the zwitterion form.

00:06:37.240 --> 00:06:39.006
And so how do you get from this form

00:06:39.006 --> 00:06:42.094
to form a peptide bond?

00:06:42.094 --> 00:06:46.342
Well, you could imagine this character over here,

00:06:46.342 --> 00:06:49.407
after giving its hydrogen protons, has an extra lone pair.

00:06:49.407 --> 00:06:53.888
So it's got one lone pair, two lone pairs,

00:06:53.888 --> 00:06:57.962
and then it's got, I'll do the extra lone pair in,

00:06:57.962 --> 00:07:00.152
I'll do the extra loan pair in purple.

00:07:00.152 --> 00:07:01.738
It's got an extra lone pair.

00:07:01.738 --> 00:07:04.547
Well, maybe it could use an extra lone pair

00:07:04.547 --> 00:07:08.123
to either grab a proton from the solution

00:07:08.123 --> 00:07:10.793
or maybe just for accounting convenience we could say,

00:07:10.793 --> 00:07:12.140
"Well, maybe just bumps in the right way

00:07:12.140 --> 00:07:16.807
"to grab this proton and then allows the nitrogen...

00:07:19.347 --> 00:07:21.612
"to take back these electrons."

00:07:21.612 --> 00:07:25.305
And if it did that, well, then you're getting,

00:07:25.305 --> 00:07:27.604
at least when you're looking at this carboxyl group

00:07:27.604 --> 00:07:29.391
and this amino group,

00:07:29.391 --> 00:07:31.319
you're going to get to the form that we just saw.

00:07:31.319 --> 00:07:32.526
If this gets a hydrogen here,

00:07:32.526 --> 00:07:34.313
this is gonna become a hydroxyl,

00:07:34.313 --> 00:07:37.588
and if this nitrogen takes back these two electrons

00:07:37.588 --> 00:07:39.980
from this pair, then it's just going to be NH2.

00:07:39.980 --> 00:07:43.625
So it's going to be, at least this part of the molecules,

00:07:43.625 --> 00:07:45.737
are gonna be just what we started with up here,

00:07:45.737 --> 00:07:47.921
and so you could imagine how you get back

00:07:47.921 --> 00:07:51.756
to the peptide linkage

00:07:51.756 --> 00:07:53.966
which we have right over here.

00:07:53.966 --> 00:07:55.421
This is the peptide linkage.

00:07:55.421 --> 00:07:59.460
And then the only difference between the resulting peptide

00:07:59.460 --> 00:08:01.759
that I have in this reaction,

00:08:01.759 --> 00:08:03.246
I guess you could say, in the previous one,

00:08:03.246 --> 00:08:05.195
is this is a zwitterion.

00:08:05.195 --> 00:08:09.212
The zwitterion form, where this carboxyl group...

00:08:10.232 --> 00:08:12.579
having donated its proton to the solution,

00:08:12.579 --> 00:08:15.575
and over here the nitrogen, this nitrogen,

00:08:15.575 --> 00:08:17.850
has taken a proton from the solution,

00:08:17.850 --> 00:08:19.383
so it has a net neutral charge,

00:08:19.383 --> 00:08:22.029
even though you do have a charge at either end.

00:08:22.029 --> 00:08:24.282
So hopefully you found that fun.


Angle measurement and circle arcs

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.410
We already know that
an angle is formed

00:00:02.410 --> 00:00:04.440
when two rays share
a common endpoint.

00:00:04.440 --> 00:00:06.650
So, for example, let's say
that this is one ray right

00:00:06.650 --> 00:00:11.000
over here, and then this is one
another ray right over here,

00:00:11.000 --> 00:00:13.264
and then they would
form an angle.

00:00:13.264 --> 00:00:14.680
And at this point
right over here,

00:00:14.680 --> 00:00:18.660
their common endpoint is called
the vertex of that angle.

00:00:18.660 --> 00:00:21.750
Now, we also know that not
all angles seem the same.

00:00:21.750 --> 00:00:23.720
For example, this
is one angle here,

00:00:23.720 --> 00:00:26.850
and then we could
have another angle

00:00:26.850 --> 00:00:31.733
that looks something like this.

00:00:34.249 --> 00:00:36.040
And viewed this way,
it looks like this one

00:00:36.040 --> 00:00:39.410
is much more open.

00:00:39.410 --> 00:00:43.070
So I'll say more open.

00:00:43.070 --> 00:00:46.800
And this one right over
here seems less open.

00:00:46.800 --> 00:00:50.049
So to avoid having to just say,
oh, more open and less open

00:00:50.049 --> 00:00:52.340
and actually becoming a little
bit more exact about it,

00:00:52.340 --> 00:00:55.830
we'd actually want to
measure how open an angle is,

00:00:55.830 --> 00:00:59.330
or we'd want to have a
measure of the angle.

00:00:59.330 --> 00:01:01.590
Now, the most typical way
that angles are measured,

00:01:01.590 --> 00:01:04.610
there's actually two major
ways of that they're measured.

00:01:04.610 --> 00:01:07.570
The most typical
unit is in degrees,

00:01:07.570 --> 00:01:09.290
but later on in
high school, you'll

00:01:09.290 --> 00:01:13.610
also see the unit of radians
being used, especially when you

00:01:13.610 --> 00:01:15.340
learn trigonometry.

00:01:15.340 --> 00:01:18.749
But the degrees convention
really comes from a circle.

00:01:18.749 --> 00:01:20.790
So let's draw ourselves
a circle right over here,

00:01:20.790 --> 00:01:22.830
so that's a circle.

00:01:22.830 --> 00:01:26.250
And the convention is that--
when I say convention,

00:01:26.250 --> 00:01:29.310
it's just kind of what
everyone has been doing.

00:01:29.310 --> 00:01:33.370
The convention is that you
have 360 degrees in a circle.

00:01:33.370 --> 00:01:34.720
So let me explain that.

00:01:34.720 --> 00:01:36.340
So if that's the
center of the circle,

00:01:36.340 --> 00:01:43.070
and if we make this ray our
starting point or one side

00:01:43.070 --> 00:01:48.600
of our angle, if you go all
the way around the circle,

00:01:48.600 --> 00:01:51.505
that represents 360 degrees.

00:01:54.390 --> 00:01:57.810
And the notation is 360, and
then this little superscript

00:01:57.810 --> 00:01:59.980
circle represents degrees.

00:01:59.980 --> 00:02:03.090
This could be read
as 360 degrees.

00:02:03.090 --> 00:02:07.500
Now, you might be saying, where
did this 360 number come from?

00:02:07.500 --> 00:02:11.790
And no one knows for sure,
but there's hints in history,

00:02:11.790 --> 00:02:15.310
and there's hints in just the
way that the universe works,

00:02:15.310 --> 00:02:17.780
or at least the Earth's
rotation around the sun.

00:02:17.780 --> 00:02:19.650
You might recognize
or you might already

00:02:19.650 --> 00:02:25.520
realize that there are 365
days in a non-leap year, 366

00:02:25.520 --> 00:02:26.810
in a leap year.

00:02:26.810 --> 00:02:29.420
And so you can imagine ancient
astronomers might have said,

00:02:29.420 --> 00:02:38.780
well, you know, that's
pretty close to 360.

00:02:38.780 --> 00:02:40.940
And in fact, several
ancient calendars,

00:02:40.940 --> 00:02:43.010
including the Persians
and the Mayans,

00:02:43.010 --> 00:02:46.360
had 360 days in their year.

00:02:46.360 --> 00:02:49.730
And 360 is also a much
neater number than 365.

00:02:49.730 --> 00:02:52.100
It has many, many more factors.

00:02:52.100 --> 00:02:55.570
It's another way of saying it's
divisible by a bunch of things.

00:02:55.570 --> 00:02:58.290
But anyway, this has just been
the convention, once again,

00:02:58.290 --> 00:03:00.950
what history has handed
us, that a circle is

00:03:00.950 --> 00:03:03.800
viewed to have 360 degrees.

00:03:03.800 --> 00:03:06.070
And so one way we
could measure an angle

00:03:06.070 --> 00:03:09.090
is you could put one of the
rays of an angle right over here

00:03:09.090 --> 00:03:14.220
at this part of the circle, and
then the other ray of the angle

00:03:14.220 --> 00:03:17.200
will look something like this.

00:03:17.200 --> 00:03:24.580
And then the fraction of
the circle circumference

00:03:24.580 --> 00:03:28.260
that is intersected by these two
rays, the measure of this angle

00:03:28.260 --> 00:03:31.010
would be that
fraction of degrees.

00:03:31.010 --> 00:03:38.900
So, for example, let's say that
this length right over here

00:03:38.900 --> 00:03:44.253
is 1/6 of the circle's
circumference.

00:03:49.080 --> 00:03:52.260
So it's 1/6 of the
way around the circle.

00:03:52.260 --> 00:03:54.290
Then this angle
right over here is

00:03:54.290 --> 00:03:58.010
going to be 1/6 of 360 degrees.

00:03:58.010 --> 00:04:01.200
So in this case, this
would be 60 degrees.

00:04:01.200 --> 00:04:02.920
I could do another example.

00:04:02.920 --> 00:04:07.970
So let's say I had a circle like
this, and I'll draw an angle.

00:04:07.970 --> 00:04:10.540
I'll put the vertex at
the center of the angle.

00:04:10.540 --> 00:04:12.330
I'll put one of the
rays right over here.

00:04:12.330 --> 00:04:15.100
You could consider
that to be 0 degrees.

00:04:15.100 --> 00:04:19.790
Or if the other ray was also
here, it would be 0 degrees.

00:04:19.790 --> 00:04:23.540
And then I'll make the
other ray of this angle,

00:04:23.540 --> 00:04:25.910
let's say it went straight up.

00:04:25.910 --> 00:04:29.490
Let's say it went
straight up like this.

00:04:29.490 --> 00:04:33.390
Well, in this
situation, the arc that

00:04:33.390 --> 00:04:37.110
connects these two
endpoints just like this,

00:04:37.110 --> 00:04:40.940
this represents 1/4 of the
circumference of the circle.

00:04:40.940 --> 00:04:44.916
This is, right over here,
1/4 of the circumference.

00:04:49.760 --> 00:04:55.400
So this angle right over here is
going to be 1/4 of 360 degrees.

00:04:55.400 --> 00:05:00.210
360 degrees divided by 4
is going to be 90 degrees.

00:05:00.210 --> 00:05:03.680
At an angle like this, one where
one ray is straight up and down

00:05:03.680 --> 00:05:07.200
and the other one goes to
the right/left direction,

00:05:07.200 --> 00:05:11.760
we would say these two
rays are perpendicular,

00:05:11.760 --> 00:05:13.940
or we would call
this a right angle.

00:05:13.940 --> 00:05:16.650
And the way that we
oftentimes will denote that

00:05:16.650 --> 00:05:19.360
is by a symbol like this.

00:05:19.360 --> 00:05:22.930
But this literally
means a 90-degree angle.

00:05:22.930 --> 00:05:24.355
Let's do one more example.

00:05:24.355 --> 00:05:25.980
Let's do one more
example of this, just

00:05:25.980 --> 00:05:28.550
to make sure that we
understand what's going on.

00:05:28.550 --> 00:05:30.810
Actually, at least
one more example.

00:05:30.810 --> 00:05:32.930
Maybe one more if we have time.

00:05:32.930 --> 00:05:35.300
So let's say that we have an
angle that looks like this.

00:05:35.300 --> 00:05:38.750
Once more, I'm going
to put its vertex

00:05:38.750 --> 00:05:40.150
at the center of the circle.

00:05:40.150 --> 00:05:42.020
That's one ray of the angle.

00:05:42.020 --> 00:05:45.110
And let's say that
this is the other ray.

00:05:45.110 --> 00:05:48.110
This right over here is
the other ray of the angle.

00:05:48.110 --> 00:05:50.300
I encourage you to
pause this video

00:05:50.300 --> 00:05:55.710
and try to figure out what
the measure of this angle

00:05:55.710 --> 00:05:57.900
right over here is.

00:05:57.900 --> 00:06:00.400
Well, let's think about where
the rays intersect the circle.

00:06:00.400 --> 00:06:02.090
They intersect there and there.

00:06:02.090 --> 00:06:04.490
The arc that connects
them on the circle

00:06:04.490 --> 00:06:06.380
is that arc right over there.

00:06:06.380 --> 00:06:09.850
That is literally half of the
circumference of the circle.

00:06:09.850 --> 00:06:13.730
That is half of the
circumference, half of the way

00:06:13.730 --> 00:06:18.010
around of the circle,
circumference of the circle.

00:06:18.010 --> 00:06:21.570
So this angle is going to
be half of 360 degrees.

00:06:21.570 --> 00:06:26.006
And half of 360 is 180 degrees.

00:06:26.006 --> 00:06:27.880
And when you view it
this way, these two rays

00:06:27.880 --> 00:06:28.970
share a common endpoint.

00:06:28.970 --> 00:06:32.572
And together, they're
really forming a line here.

00:06:32.572 --> 00:06:34.030
And let's just do
one more example,

00:06:34.030 --> 00:06:36.716
because I said I would.

00:06:36.716 --> 00:06:39.580
Let me paste another circle.

00:06:39.580 --> 00:06:42.100
Let me draw another angle.

00:06:42.100 --> 00:06:44.040
Let me draw another angle.

00:06:44.040 --> 00:06:46.540
So let's say that's
one ray of the angle,

00:06:46.540 --> 00:06:49.360
and this is the other ray.

00:06:49.360 --> 00:06:52.720
This is the other ray of
the angle right over here.

00:06:52.720 --> 00:06:53.221
And we care.

00:06:53.221 --> 00:06:55.053
There's actually two
angles that are formed.

00:06:55.053 --> 00:06:57.290
There's actually two angles
formed in all of these.

00:06:57.290 --> 00:06:59.330
There's one angle that's
formed right over here,

00:06:59.330 --> 00:07:01.970
and you might recognize that
to be a 90-degree angle.

00:07:01.970 --> 00:07:04.580
But what we really care
about in this example

00:07:04.580 --> 00:07:07.180
is this angle right over here.

00:07:07.180 --> 00:07:10.690
So once again, where does
it intersect the circle?

00:07:10.690 --> 00:07:13.580
We care about this
arc right over here,

00:07:13.580 --> 00:07:17.720
because that's the
arc that corresponds

00:07:17.720 --> 00:07:19.670
to this angle right over here.

00:07:19.670 --> 00:07:22.770
And it looks like we've
gone 3/4 around the circle.

00:07:22.770 --> 00:07:28.510
So this angle is going
to be 3/4 of 360 degrees.

00:07:28.510 --> 00:07:32.340
1/4 of 360 degrees is
90, so three of those

00:07:32.340 --> 00:07:36.142
is going to be 270 degrees.


Constructing circumscribing circle

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.360
Construct a circle
circumscribing the triangle.

00:00:03.360 --> 00:00:07.180
So that would be a circle that
touches the vertices, the three

00:00:07.180 --> 00:00:09.320
vertices of this triangle.

00:00:09.320 --> 00:00:13.020
So we can construct it using
a compass and a straight edge,

00:00:13.020 --> 00:00:15.640
or a virtual compass and
a virtual straight edge.

00:00:15.640 --> 00:00:18.480
So what we want to do is center
the circle at the perpendicular

00:00:18.480 --> 00:00:21.230
bisectors of the sides,
or sometimes that's

00:00:21.230 --> 00:00:24.710
called the circumcenter
of this triangle.

00:00:24.710 --> 00:00:26.790
So let's do that.

00:00:26.790 --> 00:00:29.660
And so let's think
about-- let's try

00:00:29.660 --> 00:00:32.500
to construct where
the perpendicular

00:00:32.500 --> 00:00:34.410
bisectors of the sides are.

00:00:34.410 --> 00:00:38.370
So let me put a
circle right over here

00:00:38.370 --> 00:00:41.400
whose radius is longer than
this side right over here.

00:00:41.400 --> 00:00:44.980
Now let me get one
that has the same size.

00:00:44.980 --> 00:00:48.730
So let me make it the same
size as the one I just did.

00:00:48.730 --> 00:00:51.320
And let me put it
right over here.

00:00:51.320 --> 00:00:55.140
And this allows us to construct
a perpendicular bisector.

00:00:55.140 --> 00:01:03.350
If I go through that point and
this point right over here,

00:01:03.350 --> 00:01:06.100
this bisects this
side over here,

00:01:06.100 --> 00:01:07.810
and it's at a right angle.

00:01:07.810 --> 00:01:10.000
So now let's do that
for the other sides.

00:01:10.000 --> 00:01:13.170
So if I move this over
here-- and I really just

00:01:13.170 --> 00:01:15.130
have to do it for one
of the other sides,

00:01:15.130 --> 00:01:17.520
because the intersection
of two lines

00:01:17.520 --> 00:01:18.780
is going to give me a point.

00:01:18.780 --> 00:01:21.260
So I can do it for this
side right over here.

00:01:21.260 --> 00:01:23.690
Let me scroll down so you
can see a little bit clearer.

00:01:23.690 --> 00:01:26.740
So let me add another
straight edge right over here.

00:01:26.740 --> 00:01:28.740
So I'm going to go
through that point,

00:01:28.740 --> 00:01:31.340
and I'm going to go
through this point.

00:01:31.340 --> 00:01:37.310
So that's the
perpendicular bisector

00:01:37.310 --> 00:01:39.540
of this side right over here.

00:01:39.540 --> 00:01:41.829
And I could do the
third side and it

00:01:41.829 --> 00:01:43.120
should intersect at that point.

00:01:43.120 --> 00:01:46.245
I'm not ultra, ultra precise,
but I'm close enough.

00:01:46.245 --> 00:01:48.370
And now I just have to
center one of these circles.

00:01:48.370 --> 00:01:50.310
Let me move one of these away.

00:01:50.310 --> 00:01:52.630
So let me just get
rid of this one.

00:01:52.630 --> 00:01:57.020
And I just have to move this
circle to the circumcenter

00:01:57.020 --> 00:02:01.710
and adjust its radius so that
it gets pretty close to touching

00:02:01.710 --> 00:02:05.961
the three sides, the three
vertices of this triangle.

00:02:05.961 --> 00:02:07.210
It doesn't have to be perfect.

00:02:07.210 --> 00:02:09.479
I think this exercise has
some margin for error.

00:02:09.479 --> 00:02:12.130
But they really want
to see that you've

00:02:12.130 --> 00:02:14.790
made an attempt at
drawing the perpendicular

00:02:14.790 --> 00:02:18.180
bisectors of the sides,
to find the circumcenter,

00:02:18.180 --> 00:02:21.180
and then you put a
circle right over there.


Constructing circle inscribing triangle

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.200
Construct a circle
inscribing the triangle.

00:00:03.200 --> 00:00:06.870
So this would be a circle that's
inside this triangle, where

00:00:06.870 --> 00:00:09.100
each of the sides
of the triangle

00:00:09.100 --> 00:00:11.550
are tangents to the circle.

00:00:11.550 --> 00:00:15.560
And probably the
easiest way to think

00:00:15.560 --> 00:00:18.830
about it is the center
of that circle is going

00:00:18.830 --> 00:00:21.022
to be at the incenter
of the triangle.

00:00:21.022 --> 00:00:22.730
Now what is the incenter
of the triangle?

00:00:22.730 --> 00:00:26.250
The incenter of the
triangle is the intersection

00:00:26.250 --> 00:00:28.580
of the angle bisectors.

00:00:28.580 --> 00:00:33.410
So if I were to make a
line that perfectly splits

00:00:33.410 --> 00:00:36.400
an angle in two-- so I'm
eyeballing it right over here--

00:00:36.400 --> 00:00:38.140
this would be an angle bisector.

00:00:38.140 --> 00:00:40.910
But to be a little bit more
precise about angle bisectors,

00:00:40.910 --> 00:00:42.540
I could actually use a compass.

00:00:42.540 --> 00:00:44.850
So let me make this
a little bit smaller.

00:00:44.850 --> 00:00:47.130
And what I can do
is I could put this,

00:00:47.130 --> 00:00:51.990
the center of this circle, on
one of the sides of this angle

00:00:51.990 --> 00:00:52.920
right over here.

00:00:52.920 --> 00:00:55.200
Now let me get another circle.

00:00:55.200 --> 00:00:57.520
And I want to make
it the same size.

00:00:57.520 --> 00:00:59.050
So let me center it there.

00:00:59.050 --> 00:01:01.590
I want to make it
the exact same size.

00:01:01.590 --> 00:01:03.600
And now let me put
it on the other one,

00:01:03.600 --> 00:01:06.200
on the other side of this angle.

00:01:06.200 --> 00:01:08.212
I'll put it right over here.

00:01:08.212 --> 00:01:10.420
And I want to put it so that
the center of the circle

00:01:10.420 --> 00:01:12.220
is on the other
side of the angle,

00:01:12.220 --> 00:01:14.400
and that the circle
itself, or the vertex,

00:01:14.400 --> 00:01:16.160
sits on the circle itself.

00:01:16.160 --> 00:01:18.020
And what this does
is I can now look

00:01:18.020 --> 00:01:20.200
at the intersection of
this point, the vertex,

00:01:20.200 --> 00:01:24.120
and this point, and that's
going to be the angle bisector.

00:01:24.120 --> 00:01:25.970
So let me go-- I'm going
to go through there

00:01:25.970 --> 00:01:29.030
and I'm going to
go through there.

00:01:29.030 --> 00:01:33.550
Now let me move these
circles over to here,

00:01:33.550 --> 00:01:37.270
so I can take the angle
bisector of this side as well.

00:01:37.270 --> 00:01:39.260
So I can put this one over here.

00:01:39.260 --> 00:01:41.360
And I could put this
one-- let's see,

00:01:41.360 --> 00:01:43.220
I want to be on the
side of the angle.

00:01:43.220 --> 00:01:47.940
And I want the circle to go
right through the vertex.

00:01:47.940 --> 00:01:50.070
Now let me add another
straight edge here.

00:01:50.070 --> 00:01:52.280
So I want to go
through this point

00:01:52.280 --> 00:01:56.220
and I want to
bisect the angle, go

00:01:56.220 --> 00:02:00.105
right through the other point
of intersection of these two

00:02:00.105 --> 00:02:02.572
circles.

00:02:02.572 --> 00:02:04.530
Now let me get rid of
one of these two circles.

00:02:04.530 --> 00:02:06.250
I don't need that anymore.

00:02:06.250 --> 00:02:08.530
And let me use this one
to actually construct

00:02:08.530 --> 00:02:10.651
the circle inscribing
the triangle.

00:02:10.651 --> 00:02:12.900
So I'm going to put it at
the center right over there.

00:02:12.900 --> 00:02:14.358
Actually, this
one's already pretty

00:02:14.358 --> 00:02:16.640
close in terms of dimensions.

00:02:16.640 --> 00:02:20.310
And with this tool, you don't
have to be 100% precise.

00:02:20.310 --> 00:02:25.210
It has some margin for error.

00:02:25.210 --> 00:02:27.060
And so let's just go with this.

00:02:27.060 --> 00:02:28.510
This actually
should be touching.

00:02:28.510 --> 00:02:29.850
But this has some
margin for error.

00:02:29.850 --> 00:02:31.266
Let's see if this
was good enough.

00:02:31.266 --> 00:02:32.700
It was.


Constructing regular hexagon inscribed in circle

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.942
Construct a regular hexagon
inscribed inside the circle.

00:00:04.942 --> 00:00:06.400
So what I'm going
to do, first, I'm

00:00:06.400 --> 00:00:08.169
going to draw a
diameter of the circle.

00:00:08.169 --> 00:00:10.210
And actually, I'm going
to go beyond the diameter

00:00:10.210 --> 00:00:10.793
of the circle.

00:00:10.793 --> 00:00:12.730
I'm just going to
draw a line that

00:00:12.730 --> 00:00:15.930
goes through the center of the
circle and just keeps on going.

00:00:15.930 --> 00:00:20.510
And I'm going to make it flat,
so it goes directly through,

00:00:20.510 --> 00:00:22.410
so this one right over
here, goes directly

00:00:22.410 --> 00:00:23.621
through the center.

00:00:23.621 --> 00:00:25.120
And now, what I'm
going to do is I'm

00:00:25.120 --> 00:00:26.870
going to construct
a circle that's

00:00:26.870 --> 00:00:29.350
the exact same dimensions of
this circle that's already

00:00:29.350 --> 00:00:29.950
been drawn.

00:00:29.950 --> 00:00:33.400
So let me put this
one right over here.

00:00:33.400 --> 00:00:36.952
And let me make it
the same dimensions.

00:00:36.952 --> 00:00:39.410
And now, what I'm going to do
is I'm going to move it over,

00:00:39.410 --> 00:00:41.970
so that this new
circle intersects

00:00:41.970 --> 00:00:43.430
the center of the old circle.

00:00:43.430 --> 00:00:44.960
And these circles
are the same size.

00:00:44.960 --> 00:00:48.360
So notice, this center
intersects the old circle.

00:00:48.360 --> 00:00:50.690
And the new circle
itself intersects

00:00:50.690 --> 00:00:52.550
the center of the old circle.

00:00:52.550 --> 00:00:54.630
Now, the reason why
this is interesting,

00:00:54.630 --> 00:00:56.630
we already know that this
distance, the distance

00:00:56.630 --> 00:01:00.740
between these two centers,
this is equal to a radius.

00:01:00.740 --> 00:01:04.989
We also know-- we have a
straight edge-- we also

00:01:04.989 --> 00:01:10.200
know that this distance right
over here is equal to a radius.

00:01:10.200 --> 00:01:13.570
It's equal to the radius of
our new circle right over here.

00:01:13.570 --> 00:01:18.340
We also know that this
distance right over here

00:01:18.340 --> 00:01:22.274
is equal to a radius
of our old circle.

00:01:22.274 --> 00:01:23.940
And that they both
have the same radius.

00:01:23.940 --> 00:01:26.460
So this is a radius
between these two points.

00:01:26.460 --> 00:01:28.310
Between these two
points is a radius.

00:01:28.310 --> 00:01:30.690
And then, in between these
two points is a radius.

00:01:30.690 --> 00:01:34.034
So now, I have constructed
an equilateral triangle.

00:01:34.034 --> 00:01:36.200
And essentially, you have
to just do this six times.

00:01:36.200 --> 00:01:38.650
And then I'm going to
have a hexagon inscribed

00:01:38.650 --> 00:01:39.400
inside the circle.

00:01:39.400 --> 00:01:40.910
Let me do it again.

00:01:40.910 --> 00:01:45.360
So we'll go from here to here.

00:01:45.360 --> 00:01:48.010
This is a radius of
my new circle, which

00:01:48.010 --> 00:01:51.215
is the same as the
radius of the old circle.

00:01:51.215 --> 00:01:55.848
And I could go
from here to here.

00:01:55.848 --> 00:01:57.306
That's the radius
of my old circle.

00:01:57.306 --> 00:01:58.930
So I have another
equilateral triangle.

00:01:58.930 --> 00:02:00.932
Radius, radius, radius.

00:02:00.932 --> 00:02:02.140
Another equilateral triangle.

00:02:02.140 --> 00:02:05.430
I've just got to do this, I
have to do this four more times.

00:02:05.430 --> 00:02:07.461
So let me go to my original.

00:02:07.461 --> 00:02:09.169
Let's see, let me make
sure I can-- well,

00:02:09.169 --> 00:02:11.418
it's actually going to be
hard for me to-- let me just

00:02:11.418 --> 00:02:14.750
add another circle here,
to do it on the other side.

00:02:14.750 --> 00:02:19.300
So if I put the
center of it right-- I

00:02:19.300 --> 00:02:22.470
want to move it a
little bit, right over--

00:02:22.470 --> 00:02:24.480
I want to make it the same size.

00:02:24.480 --> 00:02:27.310
So oh, that's close enough.

00:02:27.310 --> 00:02:32.820
And let's see, that
looks pretty close.

00:02:32.820 --> 00:02:34.010
That's the same size.

00:02:34.010 --> 00:02:35.450
Now, let me move it over here.

00:02:35.450 --> 00:02:40.350
Now, I want this center to be
on the circle, right like that.

00:02:40.350 --> 00:02:42.815
And now, I'm ready to draw some
more equilateral triangles.

00:02:46.020 --> 00:02:48.370
And really, I don't have to
even draw the inside of it.

00:02:48.370 --> 00:02:51.900
Now, I see my six
vertices for my hexagon,

00:02:51.900 --> 00:02:55.760
here, here, here,
here, here, here.

00:02:55.760 --> 00:02:58.980
And I think you're satisfied
now that you could break this up

00:02:58.980 --> 00:03:01.320
into six equilateral triangles.

00:03:01.320 --> 00:03:02.740
So let's do that.

00:03:02.740 --> 00:03:06.670
So this would be the base of one
of those equilateral triangles.

00:03:06.670 --> 00:03:08.890
And actually, let me move these.

00:03:08.890 --> 00:03:10.810
Let me move this
one out of the way.

00:03:10.810 --> 00:03:12.610
I can move this one
right over here.

00:03:12.610 --> 00:03:15.930
Because I really just care
about the hexagon itself.

00:03:15.930 --> 00:03:17.887
And I can move this
right over here.

00:03:17.887 --> 00:03:20.220
But we know that these are
all the lengths of the radius

00:03:20.220 --> 00:03:21.230
anyway.

00:03:21.230 --> 00:03:23.650
Actually, I'm not even having
to change the length there.

00:03:23.650 --> 00:03:26.590
Then, I have to just connect
one more right down here.

00:03:26.590 --> 00:03:30.010
So let me add another straight
edge, connect those two points.

00:03:30.010 --> 00:03:31.230
And I would have done it.

00:03:31.230 --> 00:03:38.930
I would have constructed my
regular hexagon inscribed

00:03:38.930 --> 00:03:40.750
in the circle.


Constructing equilateral triangle inscribed in circle

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.520
Construct an
equilateral triangle

00:00:02.520 --> 00:00:05.230
inscribed inside the circle.

00:00:05.230 --> 00:00:07.505
So let me construct
a circle that

00:00:07.505 --> 00:00:11.040
has the exact same dimensions
as our original circle.

00:00:11.040 --> 00:00:12.270
Looks pretty good.

00:00:12.270 --> 00:00:13.670
And now, let me
move this center,

00:00:13.670 --> 00:00:15.490
so it sits on our
original circle.

00:00:15.490 --> 00:00:17.180
So they now sit on each other.

00:00:17.180 --> 00:00:20.340
Or their centers now
sit on each other.

00:00:20.340 --> 00:00:24.800
So I can make it, and
that looks pretty good.

00:00:24.800 --> 00:00:26.780
And now, let's think
about something.

00:00:26.780 --> 00:00:31.290
If I were to draw this
segment right over here,

00:00:31.290 --> 00:00:34.237
this, of course, has the
length of the radius.

00:00:34.237 --> 00:00:35.320
Now, let's do another one.

00:00:35.320 --> 00:00:37.120
And that's either
of their radii,

00:00:37.120 --> 00:00:38.990
because they have
the same length.

00:00:38.990 --> 00:00:44.070
Now, let's just center
this at our new circle

00:00:44.070 --> 00:00:45.275
and take it out here.

00:00:45.275 --> 00:00:47.650
Now, this is equal to the
radius of the new circle, which

00:00:47.650 --> 00:00:49.360
is the same as the
radius of the old circle.

00:00:49.360 --> 00:00:51.276
It's going to be the
same as this length here.

00:00:51.276 --> 00:00:55.010
So these two segments
have the same length.

00:00:55.010 --> 00:01:00.420
Now, if I were to connect
that point to that point,

00:01:00.420 --> 00:01:02.900
this is a radius of
our original circle.

00:01:02.900 --> 00:01:05.510
And so it's going to have
the same length as these two.

00:01:05.510 --> 00:01:08.600
So this right over
here, I have constructed

00:01:08.600 --> 00:01:11.960
an equilateral triangle.

00:01:11.960 --> 00:01:15.490
Now, why is this at all useful?

00:01:15.490 --> 00:01:18.280
Well, we know that the angles
in an equilateral triangle

00:01:18.280 --> 00:01:19.090
are 60 degrees.

00:01:19.090 --> 00:01:23.640
So we know that this angle
right over here is 60 degrees.

00:01:23.640 --> 00:01:29.130
Now, why is this being
60 degrees interesting?

00:01:29.130 --> 00:01:33.460
Well, imagine if we constructed
another triangle out here, just

00:01:33.460 --> 00:01:36.630
symmetrically, but kind of
flipped down just like this.

00:01:36.630 --> 00:01:38.360
Well, the same argument,
this angle right

00:01:38.360 --> 00:01:40.500
over here between
these two edges,

00:01:40.500 --> 00:01:42.440
this is also going
to be 60 degrees.

00:01:42.440 --> 00:01:47.200
So this entire interior
angle, if we add those two up,

00:01:47.200 --> 00:01:49.880
are going to be 120 degrees.

00:01:49.880 --> 00:01:51.550
Now, why is that interesting?

00:01:51.550 --> 00:01:54.860
Well, if this interior
angle is 120 degrees,

00:01:54.860 --> 00:01:59.420
then that means that this arc
right over here is 120 degrees.

00:01:59.420 --> 00:02:03.380
Or it's a third of the
way around the triangle.

00:02:03.380 --> 00:02:07.410
This right over here is a third
of the way around the triangle.

00:02:07.410 --> 00:02:09.810
Since that's a third of the
way around the triangle,

00:02:09.810 --> 00:02:17.390
if I were to connect these
two dots, that is going to be,

00:02:17.390 --> 00:02:20.110
this right over
here is going to be

00:02:20.110 --> 00:02:24.020
a side of our
equilateral triangle.

00:02:24.020 --> 00:02:27.400
This right over here,
it's secant to an arc

00:02:27.400 --> 00:02:29.919
that is 1/3 of
the entire circle.

00:02:29.919 --> 00:02:31.210
And now, I can keep doing this.

00:02:31.210 --> 00:02:36.040
Let's move-- I'll reuse these--
let's move our circle around.

00:02:36.040 --> 00:02:41.420
And so now, I'm going to move
my circle along the circle.

00:02:41.420 --> 00:02:45.420
And once again, I just want
to intersect these two points.

00:02:45.420 --> 00:02:47.780
And so now, let's see, I
could take one of these,

00:02:47.780 --> 00:02:51.950
take it there, take it
there, same exact argument.

00:02:51.950 --> 00:02:56.330
This angle that I
haven't fully drawn,

00:02:56.330 --> 00:03:00.190
or this arc you could
say, is 120 degrees.

00:03:00.190 --> 00:03:03.180
So this is going to be one side
of our equilateral triangle.

00:03:03.180 --> 00:03:05.960
It's secant to an
arc of 120 degrees.

00:03:05.960 --> 00:03:07.350
So let's move this around again.

00:03:07.350 --> 00:03:10.020
Actually, we don't even have
to move this around anymore.

00:03:10.020 --> 00:03:14.060
We could just connect
those last two dots.

00:03:14.060 --> 00:03:16.010
So we could just
connect this one.

00:03:16.010 --> 00:03:19.370
Actually, I just want to, let's
connect that one to that one.

00:03:19.370 --> 00:03:23.270
And just like that,
and we're done.

00:03:23.270 --> 00:03:26.992
We have constructed our
equilateral triangle.


Parallel and perpendicular lines intro

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.460
Let's think a little
bit about two terms

00:00:02.460 --> 00:00:04.360
that you'll see
throughout your geometry,

00:00:04.360 --> 00:00:06.350
and really, mathematical career.

00:00:06.350 --> 00:00:09.730
One is the idea of things
being perpendicular.

00:00:09.730 --> 00:00:12.470
And usually, people are
talking about perpendicular.

00:00:12.470 --> 00:00:22.570
Actually I'm misspelling
it-- perpendicular lines,

00:00:22.570 --> 00:00:25.260
and the idea of parallel lines.

00:00:30.690 --> 00:00:34.050
So perpendicular
lines are two lines

00:00:34.050 --> 00:00:37.360
that intersect at a right angle.

00:00:37.360 --> 00:00:39.090
So what am I talking about?

00:00:39.090 --> 00:00:42.000
So let's say that this is
one line right over here

00:00:42.000 --> 00:00:45.790
and that this is another
line right over here.

00:00:45.790 --> 00:00:49.460
We would say these two lines are
perpendicular if they intersect

00:00:49.460 --> 00:00:50.340
at a right angle.

00:00:50.340 --> 00:00:51.950
So they clearly intersect.

00:00:51.950 --> 00:00:53.950
In order for them to
intersect at a right angle,

00:00:53.950 --> 00:00:56.280
the angle formed
between these two lines

00:00:56.280 --> 00:00:58.027
needs to be 90 degrees.

00:00:58.027 --> 00:01:00.610
And if any one of these angles
is 90 degrees, the rest of them

00:01:00.610 --> 00:01:02.114
are going to be 90 degrees.

00:01:02.114 --> 00:01:04.530
So this is 90 degrees, then
these are perpendicular lines.

00:01:04.530 --> 00:01:05.980
And if that's 90
degrees, then that's

00:01:05.980 --> 00:01:08.230
going to be 90 degrees,
that's going to be 90 degrees,

00:01:08.230 --> 00:01:09.990
and that's going
to be 90 degrees.

00:01:09.990 --> 00:01:12.073
So if any of them are 90
degrees, the rest of them

00:01:12.073 --> 00:01:15.570
are 90 degrees, and we
have perpendicular lines.

00:01:15.570 --> 00:01:18.760
If you have two lines that
on a two-dimensional surface

00:01:18.760 --> 00:01:22.050
like your paper or like
the screen never intersect,

00:01:22.050 --> 00:01:24.300
they stay the same
distance apart,

00:01:24.300 --> 00:01:27.230
then we are talking
about parallel lines.

00:01:27.230 --> 00:01:33.120
So this line right over here
and this line right over here,

00:01:33.120 --> 00:01:35.900
the way I've drawn them,
are parallel lines.

00:01:35.900 --> 00:01:37.429
They aren't intersecting.

00:01:37.429 --> 00:01:39.470
They're both kind of going
in the same direction,

00:01:39.470 --> 00:01:42.670
but they're kind of shifted
versions of each other.

00:01:42.670 --> 00:01:45.210
They will never intersect
with each other.

00:01:45.210 --> 00:01:47.380
So these two are parallel.

00:01:47.380 --> 00:01:50.420
If we have two lines that,
let's say, they intersect,

00:01:50.420 --> 00:01:52.250
but they don't intersect
at a right angle,

00:01:52.250 --> 00:01:56.250
so let's say we have that
line and we have this line

00:01:56.250 --> 00:01:59.250
right over here, and they're
clearly not intersecting

00:01:59.250 --> 00:02:03.000
at a right angle, then we call
these neither perpendicular

00:02:03.000 --> 00:02:04.160
nor parallel lines.

00:02:04.160 --> 00:02:06.674
These lines just intersect.


Which minions can the wizard reach

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.140
Alyssa is playing a video game.

00:00:02.140 --> 00:00:04.140
Her character is on
a quest to vanquish

00:00:04.140 --> 00:00:07.380
an evil sorcerer and his
minions from the land.

00:00:07.380 --> 00:00:09.850
Her character is a
wizard whose spells

00:00:09.850 --> 00:00:12.310
have a range of 6 meters.

00:00:12.310 --> 00:00:14.330
The locations of
objects in the game

00:00:14.330 --> 00:00:17.850
are stored by the computer in
terms of x and y-coordinates.

00:00:17.850 --> 00:00:21.090
So 5 comma 4 is the
location of Alyssa's wizard.

00:00:21.090 --> 00:00:24.980
8 comma 7 is the location of
Minion A. 2 comma negative 1

00:00:24.980 --> 00:00:28.090
is the location of
Minion B. 9 comma 0

00:00:28.090 --> 00:00:30.884
is the location of Minion C.

00:00:30.884 --> 00:00:33.300
So what I want to do-- and I
want you to pause this video.

00:00:33.300 --> 00:00:36.310
And I want you to think about,
given that her wizard has

00:00:36.310 --> 00:00:39.270
a range of 6 meters,
which of these minions

00:00:39.270 --> 00:00:42.169
can the wizard actually reach?

00:00:42.169 --> 00:00:43.710
I'm assuming you're
given a go at it.

00:00:43.710 --> 00:00:45.367
And we just have to
remind ourselves,

00:00:45.367 --> 00:00:47.450
to figure out which of
these minions are in reach,

00:00:47.450 --> 00:00:49.241
we have to say, well,
which of these points

00:00:49.241 --> 00:00:50.380
are within 6 units?

00:00:50.380 --> 00:00:53.290
We're assuming that these units
are in meters right over here.

00:00:53.290 --> 00:00:57.979
Which of these points are
within 6 units of 5 comma 4?

00:00:57.979 --> 00:00:59.520
And to think about
that, we just have

00:00:59.520 --> 00:01:01.353
to calculate the distance
between this point

00:01:01.353 --> 00:01:03.990
and this point, that point
and that point, that point

00:01:03.990 --> 00:01:07.220
and that point, and see if
they are greater than or less

00:01:07.220 --> 00:01:09.730
than 6 meters.

00:01:09.730 --> 00:01:12.810
And how do we calculate a
distance between two points?

00:01:12.810 --> 00:01:18.190
Well if this is some point right
over here-- that's x1 comma y1.

00:01:18.190 --> 00:01:24.690
And then this is another point
right over here, x2 comma y2.

00:01:24.690 --> 00:01:28.724
And we want to calculate this
distance right over here.

00:01:28.724 --> 00:01:31.140
The distance formula comes
straight out of the Pythagorean

00:01:31.140 --> 00:01:33.150
theorem.

00:01:33.150 --> 00:01:34.910
The Pythagorean
theorem tells us,

00:01:34.910 --> 00:01:38.224
if this side right over
here is our change in y--

00:01:38.224 --> 00:01:39.640
and let's actually
just write that

00:01:39.640 --> 00:01:42.220
as the absolute value
of our change in y--

00:01:42.220 --> 00:01:44.910
and let's say that this
side right over here

00:01:44.910 --> 00:01:49.720
is the absolute value
of our change in x,

00:01:49.720 --> 00:01:54.010
the Pythagorean theorem tells us
that this one, the hypotenuse,

00:01:54.010 --> 00:01:57.370
is going to be the square
root of the sum of the squares

00:01:57.370 --> 00:02:03.465
of the two sides-- so change
in x squared plus change in y

00:02:03.465 --> 00:02:03.965
squared.

00:02:06.197 --> 00:02:08.530
You might say, hey, what
happened to the absolute value?

00:02:08.530 --> 00:02:10.863
Well when I square it, it's
going to be positive anyway.

00:02:10.863 --> 00:02:13.230
So I don't have to write
down the absolute value.

00:02:13.230 --> 00:02:15.060
So really I just
need to figure out,

00:02:15.060 --> 00:02:17.370
between each of these
two points, what

00:02:17.370 --> 00:02:18.810
is the change in x?

00:02:18.810 --> 00:02:20.320
What's the change in y?

00:02:20.320 --> 00:02:24.310
Square them, add them together,
and then take the square root.

00:02:24.310 --> 00:02:33.650
So for example, if I
were to call this P1,

00:02:33.650 --> 00:02:37.250
if I were to call
this P2, maybe we

00:02:37.250 --> 00:02:40.480
call this P3-- I want to do
them in different colors,

00:02:40.480 --> 00:02:42.610
so you can keep track
of what I'm doing.

00:02:42.610 --> 00:02:46.950
This is P3, and
let's say this is P4.

00:02:46.950 --> 00:02:53.410
So let's first think about the
distance between P1 and P2.

00:02:56.080 --> 00:02:59.400
Well, that's going to be
equal to the square root

00:02:59.400 --> 00:03:02.760
of our change in x squared.

00:03:02.760 --> 00:03:04.860
So our change in x is 3.

00:03:04.860 --> 00:03:09.870
That squared is 9, plus
our change in y squared.

00:03:09.870 --> 00:03:11.940
Our change in y is also 3.

00:03:11.940 --> 00:03:13.680
That squared is 9.

00:03:13.680 --> 00:03:16.690
So this is going to be
square root of 18, which

00:03:16.690 --> 00:03:19.440
is the same thing as
3 square roots of 2.

00:03:22.820 --> 00:03:26.580
Now is this more or less than 6?

00:03:26.580 --> 00:03:28.370
Well 3 times 2 is equal to 6.

00:03:28.370 --> 00:03:30.190
Square root of 2 is less than 2.

00:03:30.190 --> 00:03:32.200
It's 1 point something.

00:03:32.200 --> 00:03:38.290
So this right over here is
going to be less than 6.

00:03:38.290 --> 00:03:40.460
So P2 is in range.

00:03:40.460 --> 00:03:45.340
Alissa's wizard can get Minion
A. Minion A, she can attack.

00:03:45.340 --> 00:03:54.070
Now think about Minion B. So
the distance between P1 and P3

00:03:54.070 --> 00:03:58.150
is going to be equal to
the square root of-- so

00:03:58.150 --> 00:04:02.060
your change in x,
it's negative 3.

00:04:02.060 --> 00:04:05.340
Negative 3 squared
is positive 9.

00:04:05.340 --> 00:04:10.790
Our change in y, to go from 4
to negative 1, it's negative 5.

00:04:10.790 --> 00:04:15.380
That squared is
25-- so 9 plus 25,

00:04:15.380 --> 00:04:20.829
which is equal to the
square root of 34.

00:04:20.829 --> 00:04:23.590
Now is this greater
than or equal to 6?

00:04:23.590 --> 00:04:27.360
Well, the square
root of 36 is 6.

00:04:27.360 --> 00:04:29.990
So this is the square
root of a lower number.

00:04:29.990 --> 00:04:33.400
So this is going to be
less than 6 as well.

00:04:33.400 --> 00:04:37.617
So Minion B is also in reach.

00:04:37.617 --> 00:04:39.200
Now let's think about
this last point.

00:04:39.200 --> 00:04:47.580
The distance
between P1 and P4 is

00:04:47.580 --> 00:04:52.310
going to be equal to the
square root of our change

00:04:52.310 --> 00:04:54.560
in x squared-- the
change in x is 4,

00:04:54.560 --> 00:04:58.020
squared is 16-- plus
our change in y squared.

00:04:58.020 --> 00:05:00.460
Our change in y is negative 4.

00:05:00.460 --> 00:05:03.080
But you square that,
and you get another 16.

00:05:03.080 --> 00:05:06.290
So this is going to
be square root of 32,

00:05:06.290 --> 00:05:10.000
which actually we could just
leave it as square root of 32.

00:05:10.000 --> 00:05:11.830
Square root of 32
is clearly less

00:05:11.830 --> 00:05:14.710
than the square root
of 36, which is 6.

00:05:14.710 --> 00:05:17.180
So this is also going
to be less than 6.

00:05:17.180 --> 00:05:21.330
So she can get at
all of the Minions.

00:05:21.330 --> 00:05:24.230
They're all within
6 units of her.

00:05:24.230 --> 00:05:26.330
Now which of these
is the furthest away?

00:05:26.330 --> 00:05:28.139
Well actually, if we
were to write this--

00:05:28.139 --> 00:05:29.930
we simplified this,
but we could write this

00:05:29.930 --> 00:05:32.870
as the square root of 18.

00:05:32.870 --> 00:05:34.934
Square root of 18 is
clearly the smallest out

00:05:34.934 --> 00:05:36.600
of square root of 18,
square root of 32,

00:05:36.600 --> 00:05:37.930
and square root of 34.

00:05:37.930 --> 00:05:42.290
So Minion A is the closest.

00:05:42.290 --> 00:05:48.290
And Minion B, square root
of 34, is the farthest.


Slice a rectangular pyramid

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.140
So I have a three-dimensional
solid right over here.

00:00:03.140 --> 00:00:05.920
And I want to imagine
what type of a shape

00:00:05.920 --> 00:00:09.180
I would get if I were
to make a vertical cut.

00:00:09.180 --> 00:00:12.800
And just to refresh
ourselves, or give us

00:00:12.800 --> 00:00:14.980
a sense of what a
vertical cut is,

00:00:14.980 --> 00:00:18.020
imagine if this was made
out of jello or something

00:00:18.020 --> 00:00:19.260
kind of fairly soft.

00:00:19.260 --> 00:00:21.540
But it's still a
three-dimensional solid.

00:00:21.540 --> 00:00:23.160
And I were to make
a cut-- let's say

00:00:23.160 --> 00:00:25.100
I were to make a
cut right over here.

00:00:25.100 --> 00:00:29.550
So let's say I had
this big, sharp metal

00:00:29.550 --> 00:00:31.310
thing-- let me
draw it like that.

00:00:31.310 --> 00:00:33.030
So you have this big,
sharp metal thing.

00:00:33.030 --> 00:00:36.150
Let me draw it a
little bit neater.

00:00:36.150 --> 00:00:39.260
So you have this big,
sharp metal thing

00:00:39.260 --> 00:00:42.060
that I'm going to
cut right over here.

00:00:42.060 --> 00:00:44.227
So this is the thing that
I'm going to make the cut.

00:00:44.227 --> 00:00:45.643
And I'm going to
go straight down.

00:00:45.643 --> 00:00:47.810
This is a vertical cut
that we're talking about.

00:00:47.810 --> 00:00:52.140
So this is the thing that
I'm going to cut with.

00:00:52.140 --> 00:00:53.870
Let me make it big
enough so that it

00:00:53.870 --> 00:00:57.300
can capture the shape
that will result.

00:00:57.300 --> 00:01:00.719
So this thing right over
here, it's right in front.

00:01:00.719 --> 00:01:03.010
And I'm going to cut-- I'm
going to make it go straight

00:01:03.010 --> 00:01:06.300
down and cut through
this jello, or whatever

00:01:06.300 --> 00:01:10.550
you want to call it, this
rectangular pyramid of jello.

00:01:10.550 --> 00:01:13.370
And what would be the resulting
shape of the intersection

00:01:13.370 --> 00:01:17.450
between the jello and this
thing that I'm using to cut it?

00:01:17.450 --> 00:01:19.640
And now I encourage
you to pause your video

00:01:19.640 --> 00:01:21.820
and think about what the
resulting shape would be.

00:01:21.820 --> 00:01:23.528
And the shape would
be in two dimensions.

00:01:23.528 --> 00:01:27.930
Because this purple surface
is a two-dimensional--

00:01:27.930 --> 00:01:29.830
you could view it
as part of a plane.

00:01:29.830 --> 00:01:32.570
And so where this
intersects when you cut down

00:01:32.570 --> 00:01:35.880
this rectangular pyramid is
the shape we're looking for.

00:01:35.880 --> 00:01:38.290
So I encourage you to pause
the video and think about it

00:01:38.290 --> 00:01:41.590
or try to come up
with it on your own.

00:01:41.590 --> 00:01:42.780
So let's think about it.

00:01:42.780 --> 00:01:48.190
And let me draw the
rectangular pyramid again.

00:01:48.190 --> 00:01:50.602
So that's the same one.

00:01:50.602 --> 00:01:52.560
And now let me see what
it would look like once

00:01:52.560 --> 00:01:55.220
I've done my cut, once I've
brought this thing down.

00:01:58.450 --> 00:02:00.930
So then this is where I cut.

00:02:00.930 --> 00:02:02.590
So I cut it right over here.

00:02:05.210 --> 00:02:07.970
And then it'll exit the bottom.

00:02:07.970 --> 00:02:12.690
It'll cut along
this side like that,

00:02:12.690 --> 00:02:14.860
cut along that side like that.

00:02:14.860 --> 00:02:19.870
And then it'll exit the
bottom right over there.

00:02:19.870 --> 00:02:23.070
And so let me draw
my whole thing.

00:02:23.070 --> 00:02:27.150
And so once I slice it down,
it will look like this.

00:02:30.440 --> 00:02:34.470
My best shot at drawing
it-- it will look like this.

00:02:37.420 --> 00:02:38.640
This is a vertical cut.

00:02:38.640 --> 00:02:40.150
So I've brought this thing down.

00:02:40.150 --> 00:02:42.850
And now the intersection between
the thing that I'm cutting with

00:02:42.850 --> 00:02:47.650
and this pyramid is going to
be this shape right over here.

00:02:47.650 --> 00:02:49.417
It cut into the top
right over there.

00:02:49.417 --> 00:02:51.750
It would get all the way to
the bottom right over there.

00:02:51.750 --> 00:02:54.100
And along this side, it
would cut right there.

00:02:54.100 --> 00:02:56.730
And along that side, it
would cut right over there.

00:02:56.730 --> 00:02:59.260
So what's the resulting
shape in two dimensions

00:02:59.260 --> 00:03:01.970
of essentially the
intersection between the slicer

00:03:01.970 --> 00:03:03.840
and the jello?

00:03:03.840 --> 00:03:05.420
Well, it would be this thing.

00:03:05.420 --> 00:03:06.720
It would look like a trapezoid.

00:03:06.720 --> 00:03:07.970
Let me do that in a new color.

00:03:07.970 --> 00:03:12.100
I'm overusing that one color.

00:03:12.100 --> 00:03:14.460
It would look like a trapezoid.

00:03:14.460 --> 00:03:16.110
So this would be
the resulting shape.

00:03:19.630 --> 00:03:22.050
So the resulting shape
would look like this.

00:03:25.350 --> 00:03:29.350
Just like that, if you made
the cut right over there.


Volume of a cone

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:04.140
Let's think a little bit
about the volume of a cone.

00:00:04.140 --> 00:00:06.580
So a cone would have
a circular base,

00:00:06.580 --> 00:00:09.040
or I guess depends on
how you want to draw it.

00:00:09.040 --> 00:00:11.530
If you think of like a
conical hat of some kind,

00:00:11.530 --> 00:00:13.580
it would have a
circle as a base.

00:00:13.580 --> 00:00:15.820
And it would come to some point.

00:00:15.820 --> 00:00:18.850
So it looks something like that.

00:00:18.850 --> 00:00:21.561
You could consider this to
be a cone, just like that.

00:00:21.561 --> 00:00:23.310
Or you could make it
upside down if you're

00:00:23.310 --> 00:00:25.050
thinking of an ice cream cone.

00:00:25.050 --> 00:00:26.880
So it might look
something like that.

00:00:26.880 --> 00:00:28.490
That's the top of it.

00:00:28.490 --> 00:00:31.260
And then it comes
down like this.

00:00:31.260 --> 00:00:33.310
This also is those
disposable cups of water

00:00:33.310 --> 00:00:35.559
you might see at
some water coolers.

00:00:35.559 --> 00:00:37.100
And the important
things that we need

00:00:37.100 --> 00:00:40.940
to think about when we want to
know what the volume of a cone

00:00:40.940 --> 00:00:44.410
is we definitely want to
know the radius of the base.

00:00:47.420 --> 00:00:49.990
So that's the
radius of the base.

00:00:49.990 --> 00:00:52.580
Or here is the radius
of the top part.

00:00:52.580 --> 00:00:55.190
You definitely want
to know that radius.

00:00:55.190 --> 00:00:58.980
And you want to know
the height of the cone.

00:01:01.950 --> 00:01:04.060
So let's call that h.

00:01:04.060 --> 00:01:04.950
I'll write over here.

00:01:04.950 --> 00:01:08.940
You could call this
distance right over here h.

00:01:08.940 --> 00:01:12.350
And the formula for the
volume of a cone-- and it's

00:01:12.350 --> 00:01:15.310
interesting, because it's close
to the formula for the volume

00:01:15.310 --> 00:01:17.830
of a cylinder in a
very clean way, which

00:01:17.830 --> 00:01:18.874
is somewhat surprising.

00:01:18.874 --> 00:01:20.290
And that's what's
neat about a lot

00:01:20.290 --> 00:01:21.706
of this three-dimensional
geometry

00:01:21.706 --> 00:01:24.450
is that it's not as messy as
you would think it would be.

00:01:24.450 --> 00:01:28.130
It is the area of the base.

00:01:28.130 --> 00:01:30.950
Well, what's the
area of the base?

00:01:30.950 --> 00:01:35.340
The area of the base is
going to be pi r squared.

00:01:35.340 --> 00:01:41.580
It's going to be pi r
squared times the height.

00:01:41.580 --> 00:01:44.460
And if you just multiplied
the height times pi r squared,

00:01:44.460 --> 00:01:48.460
that would give you the volume
of an entire cylinder that

00:01:48.460 --> 00:01:50.340
looks something like that.

00:01:50.340 --> 00:01:54.132
So this would give
you this entire volume

00:01:54.132 --> 00:01:55.840
of the figure that
looks like this, where

00:01:55.840 --> 00:02:00.290
its center of the top is
the tip right over here.

00:02:00.290 --> 00:02:03.210
So if I just left
it as pi r squared

00:02:03.210 --> 00:02:05.460
h or h times pi r
squared, it's the volume

00:02:05.460 --> 00:02:07.990
of this entire can,
this entire cylinder.

00:02:07.990 --> 00:02:11.260
But if you just want the
cone, it's 1/3 of that.

00:02:11.260 --> 00:02:12.595
It is 1/3 of that.

00:02:12.595 --> 00:02:13.970
And that's what
I mean when I say

00:02:13.970 --> 00:02:18.000
it's surprisingly clean that
this cone right over here

00:02:18.000 --> 00:02:21.610
is 1/3 the volume of this
cylinder that is essentially--

00:02:21.610 --> 00:02:24.656
you could think of this
cylinder as bounding around it.

00:02:24.656 --> 00:02:26.030
Or if you wanted
to rewrite this,

00:02:26.030 --> 00:02:32.930
you could write this as 1/3
times pi or pi/3 times hr

00:02:32.930 --> 00:02:33.450
squared.

00:02:33.450 --> 00:02:35.310
However you want to view it.

00:02:35.310 --> 00:02:37.080
The easy way I remember it?

00:02:37.080 --> 00:02:40.470
For me, the volume of a
cylinder is very intuitive.

00:02:40.470 --> 00:02:43.080
You take the area of the base.

00:02:43.080 --> 00:02:46.210
And then you multiply
that times the height.

00:02:46.210 --> 00:02:48.960
And so the volume of a
cone is just 1/3 of that.

00:02:48.960 --> 00:02:52.585
It's just 1/3 the volume
of the bounding cylinder

00:02:52.585 --> 00:02:53.914
is one way to think about it.

00:02:53.914 --> 00:02:55.580
But let's just apply
these numbers, just

00:02:55.580 --> 00:02:57.920
to make sure that it
makes sense to us.

00:02:57.920 --> 00:03:01.029
So let's say that this is
some type of a conical glass,

00:03:01.029 --> 00:03:03.070
the types that you might
see at the water cooler.

00:03:03.070 --> 00:03:05.500
And let's say that
we're told that it

00:03:05.500 --> 00:03:12.500
holds 131 cubic
centimeters of water.

00:03:12.500 --> 00:03:17.620
And let's say that we're
told that its height right

00:03:17.620 --> 00:03:20.640
over here-- I want to do
that in a different color.

00:03:20.640 --> 00:03:26.350
We're told that the height of
this cone is 5 centimeters.

00:03:26.350 --> 00:03:28.640
And so given that,
what is roughly

00:03:28.640 --> 00:03:31.280
the radius of the
top of this glass?

00:03:31.280 --> 00:03:34.290
Let's just say to the
nearest 10th of a centimeter.

00:03:34.290 --> 00:03:36.660
Well, we just once again
have to apply the formula.

00:03:36.660 --> 00:03:41.880
The volume, which is
131 cubic centimeters,

00:03:41.880 --> 00:03:48.010
is going to be equal
to 1/3 times pi

00:03:48.010 --> 00:03:54.340
times the height, which is 5
centimeters, times the radius

00:03:54.340 --> 00:03:56.326
squared.

00:03:56.326 --> 00:03:58.200
If we wanted to solve
for the radius squared,

00:03:58.200 --> 00:04:01.090
we could just divide both
sides by all of this business.

00:04:01.090 --> 00:04:04.820
And we would get
radius squared is

00:04:04.820 --> 00:04:11.400
equal to 131 centimeters
to the third--

00:04:11.400 --> 00:04:14.030
or 131 cubic centimeters,
I should say.

00:04:14.030 --> 00:04:15.500
You divide by 1/3.

00:04:15.500 --> 00:04:18.649
That's the same thing
as multiplying by 3.

00:04:18.649 --> 00:04:22.410
And then, of course, you're
going to divide by pi.

00:04:22.410 --> 00:04:24.920
And you're going to
divide by 5 centimeters.

00:04:27.537 --> 00:04:29.120
Now let's see if we
can clean this up.

00:04:29.120 --> 00:04:31.620
Centimeters will cancel out
with one of these centimeters.

00:04:31.620 --> 00:04:33.910
So you'll just be left
with square centimeters

00:04:33.910 --> 00:04:34.860
only in the numerator.

00:04:37.405 --> 00:04:38.780
And then to solve
for r, we could

00:04:38.780 --> 00:04:40.840
take the square
root of both sides.

00:04:40.840 --> 00:04:44.690
So we could say that
r is going to be

00:04:44.690 --> 00:04:57.230
equal to the square root of--
3 times 131 is 393 over 5 pi.

00:04:57.230 --> 00:05:00.340
So that's this part
right over here.

00:05:00.340 --> 00:05:02.330
Once again, remember
we can treat units just

00:05:02.330 --> 00:05:03.610
like algebraic quantities.

00:05:03.610 --> 00:05:05.020
The square root of
centimeters squared-- well,

00:05:05.020 --> 00:05:07.370
that's just going to be
centimeters, which is nice,

00:05:07.370 --> 00:05:09.180
because we want our
units in centimeters.

00:05:09.180 --> 00:05:11.710
So let's get our calculator
out to actually calculate

00:05:11.710 --> 00:05:14.290
this messy expression.

00:05:14.290 --> 00:05:15.410
Turn it on.

00:05:15.410 --> 00:05:15.910
Let's see.

00:05:15.910 --> 00:05:30.870
Square root of 393 divided by 5
times pi is equal to 5-- well,

00:05:30.870 --> 00:05:31.630
it's pretty close.

00:05:31.630 --> 00:05:35.130
So to the nearest, it's
pretty much 5 centimeters.

00:05:35.130 --> 00:05:40.770
So our radius is approximately
equal to 5 centimeters,

00:05:40.770 --> 00:05:43.131
at least in this example.