Triangle medians and centroids (2D proof)

WEBVTT
Kind: captions
Language: en

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- [Voiceover] In the video on
triangle medians and centroids

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I did, essentially, the
proof that the centroid

00:00:05.989 --> 00:00:08.080
is 2/3 along the way of a median.

00:00:08.080 --> 00:00:10.621
I did it using a two-dimensional triangle

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in three dimensions,

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and I mentioned that I thought, at least

00:00:12.790 --> 00:00:14.962
it made the math a little simpler,

00:00:14.962 --> 00:00:16.566
but someone mentioned
that they'd be interested

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in seeing the two-dimensional
version of the proof,

00:00:18.506 --> 00:00:20.085
so why not do that?

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So let's just draw an arbitrary triangle,

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and I'll make one side of the triangle

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right along the x-axis just to make,

00:00:26.752 --> 00:00:28.795
I think it'll make the
math a little bit easier,

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maybe there's easier ways of doing it.

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So that's one side of the triangle,

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that's the other side of the triangle,

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and let's put the height of the triangle,

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let's put it on the y-axis,

00:00:42.331 --> 00:00:43.541
so this triangle right here,

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this is the y-axis

00:00:46.806 --> 00:00:48.443
that is the y-axis,

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and then this right here,

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is our x-axis.

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Now, we said that this
is an arbitrary triangle,

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so let's just make this distance

00:01:03.517 --> 00:01:05.456
equal to a,

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so this point right here, this vertex

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would be the point a comma zero,

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so that distance is a.

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Let's make this distance right here

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be equal to, I don't know,
let's make it equal to b,

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so this is the point
negative b comma zero,

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and then let's make this distance,

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this is the point y is equal to c,

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so this is a point zero comma c,

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so it's an arbitrary triangle,

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any triangle can be represented this way.

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Now, let's think about its medians.

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Let's think about its
medians and the centroid.

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I'm only going to do it
for two of the medians,

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because we know that the third median

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will also intersect at the same centroid,

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so we have a median on this,

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we have a midpoint, I should say,

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on this side of the triangle,

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and its coordinates are just going to be

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the midpoint of those two points,

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or zero plus a over two, which
would just be a over two,

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and then c plus zero over two,

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which would just be c over two.

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Let's do the same thing over here.

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The midpoint of this side right over here

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is going to be negative
b plus zero over two,

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so it's negative b over two,

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and then zero plus c over two,

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so it's just going to be c over two.

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Now that we know all the coordinates

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for the vertices and the midpoints,

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at least of these two sides,

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we can find the equations for the lines

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that the medians are part of,

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so we can find the equation for this line,

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we could find the equation
for this line in purple,

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then we can find the
equation for the other line

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that connects to these two dots,

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find their intersection,

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and we'll essentially have the
coordinates of our centroid,

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of the intersection of the medians,

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and we can do it for this one, too,

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but that would be redundant,

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because it's going to
intersect in the same point.

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So what would be the equation
for this line right over here?

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Well, our slope is going
to be change and y,

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so c over two minus zero,

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so that's just c over two,

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over change and x,

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so a over two minus minus b,

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so minus negative b, I should say,

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so it's plus b, I could
write a plus b over here,

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but just to makes the
maths involved right,

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plus 2b over two.

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I just subtracted a negative b,

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which is the same thing as adding a b,

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and adding a b is the same
thing as adding a 2b over two,

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and I did that so these
have the same denominator.

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I could add them, or I could just multiply

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the numerator and the
denominator both by two

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and I'll get the slope is equal to,

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c over a plus 2b,

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so that's the slope of
this line right over here,

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and then we know some points here.

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We could just use the point slope formula

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for the equation of a line.

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We know the point negative
b comma zero is on the line,

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so we know that the equation
of this line in purple,

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let me stay in the purple,

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is going to be y minus zero,

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is equal to x minus negative b,

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or I could say x plus b,

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times our slope,

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times c over a plus 2b.

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And obviously, this would just simplify to

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y is equal to all of
this business over here,

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so this is essentially the equation

00:04:26.528 --> 00:04:29.295
of this median right here
if the line just kept going,

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or the median is a line segment
that is part of this line.

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Let us do the same thing for
this line right over here,

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this median right over here.

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So once again, its slope is

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change and y over change and x,

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so the change and y,
c over two minus zero,

00:04:51.293 --> 00:04:55.053
so the slope here is equal
to c over two minus zero,

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which is just c over two,

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over negative b over two,

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minus a, or I could say minus 2a over two,

00:05:05.830 --> 00:05:07.822
that's the same thing as minus a,

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and do the same thing,

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multiply the numerator and
the denominator by two.

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We get

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c over

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negative b minus 2a,

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so the equation of this line
right here, the other median,

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the equation of this other
median right over here,

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well we have this point over here,

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so we could use the point slope again.

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We get y minus zero

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is equal to x minus a

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times the slope, times c over

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negative b minus 2a,

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so we have the two
equations for these lines.

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We can now use that information
to find their intersection,

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which is going to be the centroid.

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Well, to do that, we have

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both of these in terms of y, right?

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Y minus zero is the same thing as y,

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so we could just set this

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as being equal to that, so let's do that.

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We get x plus b times c,

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over a plus 2b is equal to

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x minus a times c over

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negative b minus 2a.

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Well, both sides are divisible by c,

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we can assume that c is non-zero,

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so this triangle actually exists,

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it actually exists in two dimensions,

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so we can divide both
sides by c and we get that.

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Let's see, now we can cross-multiply.

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We can multiply x plus b times
this quantity right here,

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and that's going to be equal to

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a plus 2b times this quantity over here.

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We can just distribute it,

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so it's going to be x
times all of this business,

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it's going to be x times
negative b minus 2a,

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plus b times all of this,

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so that's going to be
minus b squared minus 2ab,

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so I just multiply that times that,

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is going to equal to this times that,

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so it's going to be x
times all of this business.

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X times a plus 2b,

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minus a times that, so distribute the a,

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minus a squared

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minus 2ab.

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Let's see if we can simplify this.

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We have a negative 2ab on both sides,

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so let's just subtract that out,

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so we cancelled things out,

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let's subtract this from
both sides of the equation,

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so we'll have a minus x times a plus 2b,

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and I'll subtract that from here,

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but I'll write it a little
different, it'll simplify things,

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so this will become

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x times, I'll distribute the
negative inside the a plus 2b,

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so negative a minus 2b,

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and let's add a b squared to both sides,

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so plus b squared, plus b squared.

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I want to collect all
the x terms on one side

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and all of the constants on the other.

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This becomes on the left-hand side,

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the coefficient on x, I have

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negative b minus 2b is negative 3b

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negative 2a minus a is negative 3a,

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times x is equal to,
these guys cancel out,

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and these guys cancel out,

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is equal to b squared minus a squared.

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Let's see if we can factor,

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this already looks a little
suspicious in a good way,

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something that we should be able to solve.

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This can be factored.

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We can factor out a negative three.

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We can actually factor out a three.

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We'll get three times b minus a,

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actually, let's factor
out a negative three,

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so negative three

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times b plus a times x

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is equal to, now this is
the same thing as b plus a

00:09:03.289 --> 00:09:04.677
times b minus a.

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So we can divide both sides by b plus a,

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we have negative three times
x is equal to b minus a,

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let's just divide both
sides by negative three.

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We get x is equal to

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b minus a over negative three,

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which is the same thing as

00:09:27.513 --> 00:09:29.559
a minus b over three,

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so we have the x-coordinate
so far for our centroid.

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It is a minus b over three.

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You kind of see hints of
the solution over here.

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A minus b is this entire distance.

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It's one third of this entire distance,

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although I don't want
to jump the gun too much

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because it complicates
just the x-coordinate.

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But let's figure out the y-coordinate.

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To do that, we can just substitute back

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into one of these equations up here.

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This was the equation of this median,

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the grey median that we had done,

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so lets substitute back in.

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We have, I'll do it over here,

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y is equal to x minus a times c.

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X is this thing over here, so it's

00:10:16.772 --> 00:10:21.772
a minus b over three minus a,

00:10:22.024 --> 00:10:23.104
so I'm going to subtract,

00:10:23.104 --> 00:10:24.876
instead of just writing
a minus a over here,

00:10:24.876 --> 00:10:28.185
I'm going to write a
negative three over three,

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so minus three a over three.

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That's this minus a right over here.

00:10:32.967 --> 00:10:36.103
I just multiplied and divided it by three.

00:10:36.103 --> 00:10:39.004
It's going to be all of that times c

00:10:40.383 --> 00:10:43.946
over negative b minus 2a,

00:10:43.946 --> 00:10:45.653
so this is going to be equal to c

00:10:45.653 --> 00:10:47.185
and the numerator up here,

00:10:47.185 --> 00:10:49.553
we have an a minus 3a,

00:10:49.553 --> 00:10:51.532
so this is equal to negative b,

00:10:54.602 --> 00:10:56.158
we have a negative b there

00:10:56.158 --> 00:11:00.297
and then we have a minus 3a is minus 2a,

00:11:00.297 --> 00:11:02.909
so this becomes negative b minus 2a,

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we can factor out the 1/3,

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times c over three, so
that's what the numerator is.

00:11:10.163 --> 00:11:12.648
All of that over negative b minus 2a,

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well, that's going to cancel with that,

00:11:14.462 --> 00:11:16.450
and so our y-coordinate is going to be

00:11:16.450 --> 00:11:19.299
equal to c over three,

00:11:19.299 --> 00:11:23.423
so our y-coordinate over
here is c over three.

00:11:23.423 --> 00:11:25.348
Now, we could just use
the distance formula now.

00:11:25.348 --> 00:11:26.346
We know all the coordinates,

00:11:26.346 --> 00:11:28.499
and feel free to do it if you like,

00:11:28.499 --> 00:11:29.915
but there's a slight simplification,

00:11:29.915 --> 00:11:32.657
or at least in my mind
a slight simplification.

00:11:32.657 --> 00:11:36.204
We know that the height,

00:11:36.204 --> 00:11:37.751
we know that the y-coordinate

00:11:40.462 --> 00:11:43.538
of the centroid is c over three,

00:11:43.538 --> 00:11:47.896
so this right here, this point
over here, is c over three.

00:11:47.896 --> 00:11:50.748
We could actually use,
maybe an easier argument,

00:11:50.748 --> 00:11:52.534
because remember, our
whole point was to show

00:11:54.355 --> 00:11:58.557
that this point is 2/3 along the median

00:11:58.557 --> 00:11:59.881
away from the vertex.

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Let me draw this vertex now here.

00:12:01.885 --> 00:12:03.043
Our whole point of this video

00:12:03.043 --> 00:12:05.693
as we said this is the
midpoint of this side over here

00:12:05.693 --> 00:12:08.073
is to show that this right here

00:12:08.073 --> 00:12:09.921
is 2/3 along the way,

00:12:09.921 --> 00:12:12.515
or that this length is twice this length,

00:12:12.515 --> 00:12:16.056
or that this length is 2/3

00:12:16.056 --> 00:12:18.714
of the entire length of the median.

00:12:18.714 --> 00:12:20.386
So how can we do that?

00:12:20.386 --> 00:12:21.907
Well, the simplest way I can do that,

00:12:21.907 --> 00:12:24.216
I mean you could actually
just use the coordinates

00:12:24.216 --> 00:12:25.307
and use the distance formula,

00:12:25.307 --> 00:12:27.287
but a simpler way, just the
way that we've arranged this,

00:12:27.287 --> 00:12:29.992
is just to use an argument
of similar triangles.

00:12:35.067 --> 00:12:38.568
We know that this entire
height right here in blue is c.

00:12:42.915 --> 00:12:46.999
We know that this height right
over here is c over three.

00:12:47.927 --> 00:12:50.505
Or you could do that as 1/3 times c,

00:12:50.505 --> 00:12:54.882
so we know that this height over here

00:12:54.882 --> 00:12:59.882
we know that that distance is 2/3 c,

00:13:00.048 --> 00:13:02.672
so we can use a similar triangle argument.

00:13:02.672 --> 00:13:07.206
Let's say that this
entire thing right here

00:13:07.206 --> 00:13:09.335
is the length of the median,

00:13:09.335 --> 00:13:10.827
and let's just call that, I don't know,

00:13:10.827 --> 00:13:12.312
let's call that l,

00:13:12.312 --> 00:13:14.221
and let's say we want to figure out

00:13:14.221 --> 00:13:16.909
how far along the median
away from the vertex that is

00:13:16.909 --> 00:13:18.407
and let's just call that,

00:13:18.407 --> 00:13:19.499
let me use another,

00:13:21.060 --> 00:13:22.806
let me just call that distance,

00:13:22.806 --> 00:13:24.280
let me do this in a different color.

00:13:26.918 --> 00:13:29.515
Let's call this distance, right over here,

00:13:29.515 --> 00:13:31.420
let's see, I've already used a, b, and c,

00:13:31.420 --> 00:13:33.196
let's call that d.

00:13:33.196 --> 00:13:35.625
So if we can show that the ratio

00:13:35.625 --> 00:13:39.552
of d to l is 2/3, that this
is 2/3 of the distance,

00:13:39.552 --> 00:13:40.749
then we're done.

00:13:40.749 --> 00:13:43.077
We can do that by similar triangles.

00:13:43.077 --> 00:13:44.430
We have two similar triangles.

00:13:45.296 --> 00:13:47.036
Let me use a new color.

00:13:47.036 --> 00:13:50.055
We have this triangle,
which is kind of embedded

00:13:50.055 --> 00:13:52.310
inside of this larger
triangle right over here.

00:13:52.310 --> 00:13:55.549
They both share the same
vertex, this angle over here,

00:13:55.549 --> 00:13:58.255
they both have right angles over here,

00:13:58.255 --> 00:13:59.870
and since they have
two of the same angles,

00:13:59.870 --> 00:14:01.358
their third angle has to be the same,

00:14:01.358 --> 00:14:03.877
so they're definitely similar triangles.

00:14:03.877 --> 00:14:05.794
So we can just set up a ratio here.

00:14:05.794 --> 00:14:09.135
2/3 c, which is this
length right over here,

00:14:10.104 --> 00:14:12.809
that's just kind of the vertical side

00:14:12.809 --> 00:14:14.620
of these smaller triangles,

00:14:14.620 --> 00:14:18.670
we could write 2/3 c
over this entire length,

00:14:18.670 --> 00:14:20.522
over the length of the larger,

00:14:20.522 --> 00:14:24.118
this corresponding side of
the larger similar triangle,

00:14:24.118 --> 00:14:26.434
over c is equal to

00:14:28.087 --> 00:14:30.811
the hypotenuse of the
smaller similar triangle,

00:14:31.661 --> 00:14:36.579
d over the hypotenuse of
the longer similar triangle,

00:14:36.579 --> 00:14:37.903
or the bigger similar triangle.

00:14:37.903 --> 00:14:39.203
D over l.

00:14:39.203 --> 00:14:40.704
This is clearly just dividing

00:14:40.704 --> 00:14:42.560
the numerator and denominator by c,

00:14:42.560 --> 00:14:44.383
this clearly becomes 2/3,

00:14:44.383 --> 00:14:48.156
so the ratio of this length

00:14:48.156 --> 00:14:50.979
to this larger length is 2/3,

00:14:50.979 --> 00:14:55.498
or this is 2/3 along the median

00:14:55.498 --> 00:14:56.897
away from the vertex.

00:14:56.897 --> 00:14:57.629
So there you go,

00:14:57.629 --> 00:14:59.365
that's the two-dimensional proof

00:14:59.365 --> 00:15:03.359
that the centroid is 2/3 along the way

00:15:03.359 --> 00:15:04.916
of any median from the vertex,

00:15:04.916 --> 00:15:08.039
or 1/3 along the median,

00:15:08.039 --> 00:15:10.188
or 1/3 the distance of the median

00:15:10.188 --> 00:15:12.487
from the opposite midpoint.

00:15:12.487 --> 00:15:13.815
Anyway, hopefully you enjoyed that.


2003 AIME II problem 6

WEBVTT
Kind: captions
Language: en

00:00:00.362 --> 00:00:08.170
In triangle ABC, AB is equal to 13, BC is equal to 14, and AC is equal to 15.

00:00:08.555 --> 00:00:14.316
Let me just draw that. It looks like a pretty involved problem

00:00:17.670 --> 00:00:21.332
So it looks something like that

00:00:31.486 --> 00:00:34.381
Let's say this is triangle ABC

00:00:34.381 --> 00:00:42.203
They tell us AB is equal to 13, BC is equal to 14 and AC is equal to 15

00:00:42.203 --> 00:00:48.033
and then they tell us point G is the intersection of the medians.

00:00:48.033 --> 00:00:51.707
If you've watched the video on medians and centroids you might recognize

00:00:51.707 --> 00:00:54.721
that the intersection of the medians is just the centroid.

00:00:54.828 --> 00:00:57.981
Let me just show that here.

00:00:59.012 --> 00:01:05.153
The line from B to the midpoint of AC is a median.

00:01:05.153 --> 00:01:08.871
Let me draw that a little cleaner.

00:01:08.871 --> 00:01:10.639
So that is a median.

00:01:10.639 --> 00:01:22.239
Now the line from A to the midpoint of BC is also a median.

00:01:22.239 --> 00:01:33.576
And finally the line from C to the midpoint of AB is a median.

00:01:33.576 --> 00:01:38.685
We learned -- if you've watched the video on medians and centroids --

00:01:38.685 --> 00:01:42.133
that these three medians always intersect at a point called the centroid.

00:01:42.133 --> 00:01:50.364
They even imply that in the question. So that right there is point G.

00:01:50.364 --> 00:01:59.267
Point A', B', and C' are the iamges of A, B, and C after a 180 degree rotation about G.

00:01:59.267 --> 00:02:06.600
What is the area of the nion of the two regions enclosed by the triangles ABC and A'B'C'?

00:02:06.600 --> 00:02:11.216
Let's draw the second triangle.

00:02:11.216 --> 00:02:16.162
What we want to do is rotate each of these about G 180 degrees.

00:02:16.162 --> 00:02:23.221
So if A is here and we rotate it 180 degrees about G,

00:02:23.221 --> 00:02:28.591
it's going to end up just as far from G on that side.

00:02:28.591 --> 00:02:36.672
So A' is going to be right over there.

00:02:36.672 --> 00:02:44.067
B is this far right now, if you rotate it 180 degrees,

00:02:44.067 --> 00:02:49.157
B' will then be about that far.

00:02:49.157 --> 00:02:52.896
Same for C. If you rotate it 180 degrees,

00:02:52.896 --> 00:03:02.671
it's going to go about here. So you have C'.

00:03:02.671 --> 00:03:07.733
So if we wanted to draw A'B'C', it's going to look like this.

00:03:07.733 --> 00:03:11.295
Here's my best attempt at it.

00:03:20.018 --> 00:03:24.548
This line should be parallel to that line

00:03:24.548 --> 00:03:27.600
since it's been rotated by 180 degrees.

00:03:27.600 --> 00:03:33.438
And this line [C'B'] should be parallel to this line [BC],

00:03:33.438 --> 00:03:38.733
and this line [C'A'] should be parallel to that line [AC].

00:03:38.733 --> 00:03:42.888
It seems like it's going to be relevant.

00:03:42.888 --> 00:03:48.554
So they're asking, they want the area of the union of the two triangles.

00:03:48.554 --> 00:03:59.667
So it's the area of this star. This slightly skewed or tilted star of David

00:03:59.667 --> 00:04:05.179
I'm not doing this in real time; it took me a while before I did this problem

00:04:05.179 --> 00:04:06.924
before it jumped out at me.

00:04:06.924 --> 00:04:09.438
Kind of knowing that most of these problems don't require

00:04:09.438 --> 00:04:10.920
really hard mathematics.

00:04:10.920 --> 00:04:12.614
A lot of the time you just kind of have to see something

00:04:12.614 --> 00:04:16.667
or rearrange things and the answer jumps out.

00:04:16.667 --> 00:04:20.017
It's also the case with this one here.

00:04:20.017 --> 00:04:23.533
And the thing that jumps out (and the motivation for making this video)

00:04:23.533 --> 00:04:33.043
I showed you in the last video that if you have the median of a triangle

00:04:33.043 --> 00:04:35.900
and the centroid of the trianlge is over here,

00:04:35.900 --> 00:04:43.200
this distance [GA] is twice the distance from the centroid to the midpoint [of BC]

00:04:43.200 --> 00:04:49.715
and the same on all 3 sides.

00:04:49.715 --> 00:05:01.229
But this distance [AG] is also the same as this whole entire distance [A'G]

00:05:01.229 --> 00:05:05.701
This distance has been rotated 180 and degrees

00:05:05.701 --> 00:05:10.369
and if this distance [G-midpoint of BC] is half of that one [AG]

00:05:10.369 --> 00:05:19.133
this is exactly halfway between this point [G] and that point [A']

00:05:19.133 --> 00:05:23.895
Another way to think about it:

00:05:23.895 --> 00:05:27.308
we know that this length over here is equal to this length over here,

00:05:27.308 --> 00:05:31.976
which is equal to this length over here, which is equal to this length over here

00:05:31.976 --> 00:05:37.061
And so you start seeing, maybe, some triangles will jump out.

00:05:37.061 --> 00:05:42.634
Let's see if I can draw these triangles.

00:05:42.634 --> 00:05:49.484
I have this triangle up here.

00:05:49.484 --> 00:05:53.849
This is actually going to be a median of this triangle over here

00:05:53.849 --> 00:06:05.227
It's going to be the same as the median or equivalent line of this triangle over here

00:06:05.227 --> 00:06:10.149
It's actually going to be an altitude -- we're kind of drawing a rhombus

00:06:10.149 --> 00:06:15.699
and the diagonals of a rhombus are perpendicular.

00:06:15.699 --> 00:06:22.456
So this altitude right here is going to be the same as this altitude over here.

00:06:22.456 --> 00:06:31.442
This side is the exact same thing as this side.

00:06:31.442 --> 00:06:33.857
Let me be careful here.

00:06:33.857 --> 00:06:37.154
We already know that this side is going to be the same thing as this side

00:06:37.154 --> 00:06:41.218
We know these two sides are the same because this is acting as a median of this triangle

00:06:41.218 --> 00:06:48.718
We already know that this yellow side I'll draw two marks is the same thing

00:06:48.718 --> 00:06:51.733
as that yellow side down there.

00:06:51.733 --> 00:06:56.496
If this is the same as this, and this is the same in both triangles,

00:06:56.496 --> 00:07:06.458
we know that this side must be the same thing as this side,

00:07:06.458 --> 00:07:16.535
which also must be the same thing as this side and this side over here.

00:07:16.535 --> 00:07:20.436
Another way to think about it, this triangle is going to have the exact same area

00:07:20.436 --> 00:07:22.085
as this triangle.

00:07:22.085 --> 00:07:24.639
This triangle is going to have the same exact area as this triangle.

00:07:24.639 --> 00:07:26.288
Or, you could view it another way.

00:07:26.288 --> 00:07:35.297
This purple triangle right over here is going to be a congruent triangle to this triangle over here.

00:07:35.297 --> 00:07:39.941
All of its sides are exactly the same. They share one of them.

00:07:39.941 --> 00:07:43.981
We could use the same argument to say that this triangle right over here

00:07:43.981 --> 00:07:54.035
is completely congruent to this triangle right over there.

00:07:54.035 --> 00:07:55.359
They have all of the same sides.

00:07:55.359 --> 00:07:59.520
In fact, we even know that they are congruent to these two up here.

00:07:59.520 --> 00:08:04.815
So these are all congruent.

00:08:04.815 --> 00:08:19.949
Let's think about this triangle right over here.

00:08:19.949 --> 00:08:27.751
We know that this distance, the distance when started talking about medians,

00:08:27.751 --> 00:08:32.534
from the centroid to the base, that's the exact same distance

00:08:32.534 --> 00:08:39.129
over here , if you were to draw a parallel line.

00:08:39.129 --> 00:08:44.631
It's just that this line up here and this line over here are parallel

00:08:44.631 --> 00:08:50.841
If this is parallel to this, and they're intersecting at the same point

00:08:50.841 --> 00:08:54.652
I won't go into the depth of proving they're parallel -- it's not too hard to prove.

00:08:54.652 --> 00:09:01.652
You can show that this distance is the same as this distance.

00:09:01.652 --> 00:09:08.688
We share this magenta side right over here,

00:09:08.688 --> 00:09:13.790
so we know that this side is going to

00:09:13.790 --> 00:09:17.256
be the same as that side

00:09:17.256 --> 00:09:19.345
Or we know that this triangle is going to be

00:09:19.345 --> 00:09:21.157
the same thing as that triangle.

00:09:21.157 --> 00:09:26.242
And then we can take the mirror image.

00:09:26.242 --> 00:09:28.355
Using essentially the same argument,

00:09:28.355 --> 00:09:31.048
that's going to be the same thing as this triangle over here.

00:09:31.048 --> 00:09:33.138
And it's a similar argument to show that that's going

00:09:33.138 --> 00:09:35.832
to be the same thing as that triangle over there.

00:09:35.832 --> 00:09:42.124
And then do it on the other side.

00:09:42.124 --> 00:09:44.237
I think you see where this is going!

00:09:44.237 --> 00:09:46.735
All of these triangles are congruent.

00:09:58.976 --> 00:10:04.148
There's actually multiple ways you can make that argument.

00:10:04.148 --> 00:10:07.597
The same argument we made to show that these magenta triangles are congruent,

00:10:07.597 --> 00:10:13.750
we could make to say that this distance is exactly half of this entire distance.

00:10:13.750 --> 00:10:16.443
Same argument to tell you that this triangle's congruent

00:10:16.443 --> 00:10:17.790
to that that triangle.

00:10:17.790 --> 00:10:21.993
And that one to that one.

00:10:21.993 --> 00:10:23.200
So that entire shape --

00:10:23.200 --> 00:10:29.168
and actually I was a little handwavy about it --

00:10:29.168 --> 00:10:33.231
I think you can see it actually wouldn't be hard to do a rigorous proof.

00:10:33.231 --> 00:10:37.202
But this shape, this slanted star of david

00:10:37.202 --> 00:10:48.557
is made up of [counts to 12] congruent triangles.

00:10:48.557 --> 00:10:53.363
12 congruent triangles.

00:10:53.363 --> 00:10:58.611
Now, how many of these congruent triangles originally made up our triangle ABC?

00:10:58.611 --> 00:11:12.729
[counts to 9]

00:11:12.729 --> 00:11:17.976
So we have 12 congruent triangles in the star

00:11:17.976 --> 00:11:30.422
we had 9 congruent triangles in ABC,

00:11:30.422 --> 00:11:37.922
so the area of the star is just going to be 12/9 the area of ABC.

00:11:37.922 --> 00:11:41.219
So if we can figure out the area of ABC,

00:11:41.219 --> 00:11:44.401
we know that each triangle will have 1/9 that area.

00:11:44.401 --> 00:11:49.788
Multiplying by 12 we'll get the area of the actual star.

00:11:49.788 --> 00:11:53.990
So let's figure out the area of triangle ABC.

00:11:53.990 --> 00:12:02.814
Let me redraw the triangle.

00:12:02.814 --> 00:12:07.017
There are formulas for figuring out the areas of triangles if you know the sides.

00:12:07.017 --> 00:12:10.918
I always forget them, but I do remember the law of cosines.

00:12:10.918 --> 00:12:15.980
It helps me derive those formulas.

00:12:15.980 --> 00:12:22.876
Area of a triangle is 1/2 base * height,

00:12:22.876 --> 00:12:27.079
so if we could figure out this height,

00:12:27.079 --> 00:12:31.421
we could multiply 1/2 and 14, or multiply it by 7 and

00:12:31.421 --> 00:12:34.718
then we would be able to figure out the area of this entire triangle.

00:12:34.718 --> 00:12:36.506
How can we do that?

00:12:36.506 --> 00:12:44.421
Let's just break out a little trigonometry.

00:12:44.421 --> 00:12:48.372
Let's take this to be theta over here.

00:12:48.372 --> 00:12:50.926
The law of cosines tells us that

00:12:50.926 --> 00:12:55.732
the opposite side of the angle squared, so 15 squared

00:12:55.732 --> 00:12:59.773
is equal to the sum of the other sides squared

00:12:59.773 --> 00:13:02.327
(13 squared + 14 squared)

00:13:02.327 --> 00:13:07.598
minus 2 times the product of these two sides

00:13:07.598 --> 00:13:10.467
so times 13 times 14

00:13:10.467 --> 00:13:15.957
times the cosine of theta

00:13:15.957 --> 00:13:18.650
Now, what is the cosine of theta?

00:13:18.650 --> 00:13:22.528
Cosine is adjacent over hypotenuse,

00:13:22.528 --> 00:13:26.568
so that's this side over our hypotenuse.

00:13:26.568 --> 00:13:36.181
So let's just call this length over here x.

00:13:36.181 --> 00:13:41.545
So cosine of theta is x over 13.

00:13:41.545 --> 00:13:45.911
So if you could solve for x you could then solve for h using the Pythogorean Theorem

00:13:45.911 --> 00:13:52.203
These two characters cancel out, and we are left with

00:13:52.203 --> 00:13:56.708
15 squared is 225, 13 squared is 169

00:13:56.708 --> 00:14:07.203
14 squared -- we have 140 + 40 + 16 = 196

00:14:07.203 --> 00:14:16.840
minus 28 x

00:14:16.840 --> 00:14:22.227
So let's solve for x.

00:14:22.227 --> 00:14:28.960
We'll do a couple of things. We'll subtract 225 from both sides

00:14:28.960 --> 00:14:36.878
and I'm going to add 28x to both sides

00:14:36.878 --> 00:14:43.357
On the left hand side we are left with 28 x is equal to

00:14:43.357 --> 00:14:44.703
and we have to do a little bit of math

00:14:44.703 --> 00:15:00.145
[does arithmetic aloud] 365. Did I do that right?

00:15:00.145 --> 00:15:07.204
And from that we want to subtract 225

00:15:07.204 --> 00:15:10.663
It is going to be 140

00:15:10.663 --> 00:15:19.208
so x is going to be 140 divided by 28

00:15:19.208 --> 00:15:25.060
Let's think about it. It goes 5 times. 5 times 20 is 100,

00:15:25.060 --> 00:15:28.659
5 times 8 is 40, so 5 times 28 is 140

00:15:28.659 --> 00:15:33.860
So x in this situation is equal to 5.

00:15:33.860 --> 00:15:37.900
So we can use that information to figure out this height.

00:15:37.900 --> 00:15:38.969
We know from the pythagorean theorem that

00:15:38.969 --> 00:15:41.221
5 squared plus h squared is equal to the hypotenuse squared.

00:15:41.221 --> 00:15:47.235
It's equal to 169

00:15:47.235 --> 00:15:53.365
or h squared = 169 -25 = 144.

00:15:53.365 --> 00:15:58.032
h is equal to 12

00:15:58.032 --> 00:16:00.865
Now we can figure out the area of the triangle.

00:16:00.865 --> 00:16:06.438
The area sometimes denoted by brackets [ABC] is equal to

00:16:06.438 --> 00:16:13.311
1/2 times 14 times 12

00:16:13.311 --> 00:16:19.302
which is the same as 7 times 12, which is equal to 84

00:16:19.302 --> 00:16:26.802
That's just the area of these 9 congruent triangles.

00:16:26.802 --> 00:16:34.302
We know that the area of the whole thing will be 12/9 of that

00:16:34.302 --> 00:16:50.811
So the area of the star is going to be equal to 12/9 times 84

00:16:50.811 --> 00:17:02.050
12/9, dividing numerator and denominator by 3 is the same as 4/3

00:17:02.050 --> 00:17:06.694
3 does go into 84. How many times? Is it 28?

00:17:06.694 --> 00:17:15.517
Let me make sure. 2*3 = 6, so yup, 28 times.

00:17:15.517 --> 00:17:19.418
So this is equal to 4 times 28

00:17:19.418 --> 00:17:23.598
Which is going to be 80 plus 32

00:17:23.598 --> 00:17:28.079
which is equal to 112.

00:17:28.079 --> 00:17:32.282
Let me make sure I did that right

00:17:32.282 --> 00:17:37.553
4 times 20 is 80 plus 32 is 112.

00:17:37.553 --> 99:59:59.999
And we are done!


2003 AIME II problem 5 minor correction

WEBVTT
Kind: captions
Language: en

00:00:00.621 --> 00:00:03.048
I want to do a quick correction to the AIME problem

00:00:03.048 --> 00:00:06.432
where we cut out a wedge of a log in that problem.

00:00:06.432 --> 00:00:07.825
So we had a wedge

00:00:07.825 --> 00:00:10.887
so just to remind ourselves that we had a log

00:00:10.887 --> 00:00:13.145
and it looked like that

00:00:13.145 --> 00:00:14.916
we made one cut that was right along

00:00:14.916 --> 00:00:17.139
that was perpendicular to the axis of the log

00:00:17.139 --> 00:00:18.039
and we made another cut

00:00:18.039 --> 00:00:20.421
that had a 45 degree angle.

00:00:20.421 --> 00:00:22.082
So it was like that.

00:00:22.082 --> 00:00:23.549
And the solution to the problem

00:00:23.549 --> 00:00:25.104
we had to figure out the volume of the wedge

00:00:25.104 --> 00:00:26.564
and the big trick there was

00:00:26.564 --> 00:00:28.723
if I drew the wedge like this

00:00:28.723 --> 00:00:30.395
so let me take a side view of the wedge.

00:00:30.395 --> 00:00:34.148
So this is the side of the wedge right over here.

00:00:34.148 --> 00:00:37.314
I could shade it to show that it's kind of round.

00:00:37.314 --> 00:00:38.401
So if I were to shade it

00:00:38.401 --> 00:00:39.824
you could kind of see that it is round.

00:00:39.824 --> 00:00:41.687
So this is a side view of the wedge.

00:00:41.687 --> 00:00:44.517
We knew that this right here is a 45 degree angle.

00:00:44.517 --> 00:00:45.813
And the way that we solved the problem

00:00:45.813 --> 00:00:47.420
or the kind of trick was to say

00:00:47.420 --> 00:00:48.892
Hey, if we had another wedge like this

00:00:48.892 --> 00:00:50.644
and we just stacked it on top of this one

00:00:50.644 --> 00:00:53.714
if we just flipped it over and made it like this

00:00:53.714 --> 00:00:57.655
then we also have a 45 degree angle over here

00:00:57.655 --> 00:00:58.719
and it would make a cylinder

00:00:58.719 --> 00:01:00.694
and it's very easy

00:01:00.694 --> 00:01:02.549
to figure out the height of a cylinder.

00:01:02.549 --> 00:01:04.219
And I misspoke when I said it was lucky

00:01:04.219 --> 00:01:06.060
that we had a 45 degree angle.

00:01:06.060 --> 00:01:07.274
Because in the last video

00:01:07.274 --> 00:01:09.335
I was actually visualizing it incorrectly.

00:01:09.335 --> 00:01:10.147
Because I said if you had anything

00:01:10.147 --> 00:01:11.926
less than a 45 degree angle

00:01:11.926 --> 00:01:13.553
this wouldn't have worked out properly

00:01:13.553 --> 00:01:14.406
But I was wrong!

00:01:14.406 --> 00:01:16.136
It would have worked out properly.

00:01:16.138 --> 00:01:17.485
Let me show you.

00:01:17.485 --> 00:01:22.640
So this, let's just say whatever angle you have here.

00:01:22.640 --> 00:01:24.590
So this is going to be 90 degrees over here.

00:01:24.590 --> 00:01:27.005
Whatever angle you have over here

00:01:27.005 --> 00:01:27.980
so let's call that theta

00:01:27.980 --> 00:01:30.198
this over here, they have to add up to 180.

00:01:30.198 --> 00:01:33.364
So theta plus 90 plus this has to be 180

00:01:33.364 --> 00:01:36.569
Or theta plus this has to be 90

00:01:36.569 --> 00:01:40.315
or you could just call this 90 minus theta.

00:01:40.315 --> 00:01:43.600
Now, if I were to take the same thing

00:01:43.600 --> 00:01:46.267
if I took another kind of wedge like this

00:01:46.267 --> 00:01:48.252
and flipped it over this thing

00:01:48.252 --> 00:01:50.048
it's not this angle that's going to be right here

00:01:50.048 --> 00:01:51.093
it's this angle.

00:01:51.093 --> 00:01:52.539
Let me just draw it over here.

00:01:52.539 --> 00:01:57.723
It would look like this.

00:01:57.723 --> 00:01:59.194
Where this now is theta.

00:01:59.194 --> 00:02:00.324
This end right here

00:02:00.324 --> 00:02:03.038
is actually that end on the flipped version.

00:02:03.038 --> 00:02:04.757
And now this angle right over here

00:02:04.757 --> 00:02:06.942
is now 90 minus theta.

00:02:06.942 --> 00:02:09.674
So clearly you have 90 minus theta plus theta.

00:02:09.674 --> 00:02:11.902
So this thing is going to be a right angle

00:02:11.902 --> 00:02:13.934
no matter what.

00:02:13.934 --> 00:02:15.826
Obviously it has to be a reasonable angle

00:02:15.826 --> 00:02:16.526
but no matter

00:02:16.526 --> 00:02:18.203
assuming that its less than 90 degrees

00:02:18.203 --> 00:02:19.260
it would have worked out.

00:02:19.260 --> 00:02:20.523
You could have done the same trick.

00:02:20.523 --> 00:02:21.821
Anyway, hopefully that clarifies it

00:02:21.821 --> 00:02:22.495
and I apologize

00:02:22.495 --> 00:02:26.329
for the incorrect visualization in the last video.


Triangle medians and centroids

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.360
I want to do a quick refresher
on medians of triangles,

00:00:03.360 --> 00:00:06.440
and also explore an interesting
property of them that

00:00:06.440 --> 00:00:08.960
will be useful, I think,
in future problems.

00:00:08.960 --> 00:00:15.030
So let me just draw an
arbitrary triangle over here.

00:00:15.030 --> 00:00:16.620
Now that's good enough.

00:00:16.620 --> 00:00:18.620
Now a median of the
triangle-- and we'll

00:00:18.620 --> 00:00:20.540
see a triangle has
three of them--

00:00:20.540 --> 00:00:23.320
is just a line that
connects a vertex

00:00:23.320 --> 00:00:26.790
of the triangle with the
midpoint of the opposite side.

00:00:26.790 --> 00:00:29.660
So the opposite side's midpoint
looks right about there.

00:00:29.660 --> 00:00:31.460
This length is equal
to that length.

00:00:31.460 --> 00:00:35.280
And so this is a median.

00:00:35.280 --> 00:00:36.437
Close enough.

00:00:36.437 --> 00:00:38.020
And of course, we
have three vertices,

00:00:38.020 --> 00:00:38.930
so we'll have three medians.

00:00:38.930 --> 00:00:40.388
If we start at this
vertex, we want

00:00:40.388 --> 00:00:42.910
to go to the midpoint
of the opposite side.

00:00:42.910 --> 00:00:44.890
It looks right about there.

00:00:44.890 --> 00:00:48.480
So this blue line right
over here is another median.

00:00:48.480 --> 00:00:49.980
It's not a completely
straight line,

00:00:49.980 --> 00:00:51.188
but I think you get the idea.

00:00:51.188 --> 00:00:53.870
Then we could also do it from
this point right over here.

00:00:53.870 --> 00:00:56.920
Draw a line from this
vertex to the midpoint

00:00:56.920 --> 00:00:57.970
of the opposite side.

00:00:57.970 --> 00:01:00.350
Let's see, the midpoint of
the opposite side is there.

00:01:00.350 --> 00:01:01.810
And we draw a line.

00:01:01.810 --> 00:01:05.313
Each of these-- I could draw
a straighter line than that.

00:01:05.313 --> 00:01:06.550
Let me draw it.

00:01:09.090 --> 00:01:09.710
There.

00:01:09.710 --> 00:01:11.001
Well, I think you get the idea.

00:01:11.001 --> 00:01:13.860
These are all medians
of this triangle.

00:01:13.860 --> 00:01:19.040
And what's neat about medians
is that all three medians always

00:01:19.040 --> 00:01:20.780
intersect in one point.

00:01:20.780 --> 00:01:22.730
And that by itself is
a pretty neat property.

00:01:22.730 --> 00:01:24.490
And that one point
that they intersect in

00:01:24.490 --> 00:01:25.448
is called the centroid.

00:01:28.540 --> 00:01:30.590
And if this was actually
a physical triangle,

00:01:30.590 --> 00:01:32.170
let's say you made
it out of iron,

00:01:32.170 --> 00:01:35.717
and if you were to toss it--
well, even before you toss it,

00:01:35.717 --> 00:01:37.800
the centroid would actually
be the center of mass.

00:01:37.800 --> 00:01:40.240
So let's say this
is an iron triangle.

00:01:40.240 --> 00:01:42.930
Let's say that this right
here is an iron triangle that

00:01:42.930 --> 00:01:45.940
has its centroid
right over here,

00:01:45.940 --> 00:01:48.670
then this iron
triangle's center of mass

00:01:48.670 --> 00:01:50.570
would be where the
centroid is, assuming

00:01:50.570 --> 00:01:52.240
it has a uniform density.

00:01:52.240 --> 00:01:56.020
And if you were to throw
that iron triangle,

00:01:56.020 --> 00:01:57.910
it would rotate
around this point.

00:02:00.530 --> 00:02:03.330
Assuming that it had
some rotational motion,

00:02:03.330 --> 00:02:06.200
it would rotate
around that centroid,

00:02:06.200 --> 00:02:07.632
or around the center of mass.

00:02:07.632 --> 00:02:09.090
But anyway, the
point of this video

00:02:09.090 --> 00:02:12.360
is not to focus on physics
and throwing iron triangles.

00:02:12.360 --> 00:02:15.660
The point here is I want to show
you a neat property of medians.

00:02:15.660 --> 00:02:20.620
And the property is that
if you pick any median,

00:02:20.620 --> 00:02:23.980
the distance from the centroid
to the midpoint of the opposite

00:02:23.980 --> 00:02:25.800
side-- so this
distance-- is going

00:02:25.800 --> 00:02:27.820
to be half of this distance.

00:02:27.820 --> 00:02:29.990
So if this distance
right here is a,

00:02:29.990 --> 00:02:32.440
then this distance
right here is 2a.

00:02:32.440 --> 00:02:34.750
Or another way to think
about it is this distance

00:02:34.750 --> 00:02:37.630
is 2/3 of the length
of the entire median,

00:02:37.630 --> 00:02:40.170
and this distance
right here is 1/3

00:02:40.170 --> 00:02:42.440
of the length of
the entire median.

00:02:42.440 --> 00:02:44.090
And let's just prove
it for ourselves

00:02:44.090 --> 00:02:47.120
just so you don't have
to take things on faith.

00:02:47.120 --> 00:02:49.750
And to do that, I'll draw
an arbitrary triangle.

00:02:49.750 --> 00:02:51.340
I'll do a
two-dimensional triangle,

00:02:51.340 --> 00:02:52.756
and I'll do it in
three dimensions

00:02:52.756 --> 00:02:55.210
because at least in
my mind, it makes

00:02:55.210 --> 00:02:57.040
the math a little bit easier.

00:02:57.040 --> 00:03:01.070
In general, whenever you
have an n-dimensional figure

00:03:01.070 --> 00:03:03.230
and you embed it in
n plus 1 dimensions,

00:03:03.230 --> 00:03:06.140
it makes the math a
little bit easier.

00:03:06.140 --> 00:03:08.204
The actual tetrahedron
problem that we did,

00:03:08.204 --> 00:03:10.120
you could actually embed
it in four dimensions

00:03:10.120 --> 00:03:11.140
and it would make
the math easier.

00:03:11.140 --> 00:03:12.750
It's just much
harder to visualize,

00:03:12.750 --> 00:03:14.080
so I didn't do it that way.

00:03:14.080 --> 00:03:16.530
But let's just have
an arbitrary triangle.

00:03:16.530 --> 00:03:18.600
And let's say it has
a vertex and there,

00:03:18.600 --> 00:03:20.692
a vertex there,
and a vertex there.

00:03:20.692 --> 00:03:22.900
So I'm not making any
assumptions about the triangle.

00:03:22.900 --> 00:03:25.316
I'm not saying it's isosceles,
or equilateral or anything.

00:03:25.316 --> 00:03:27.150
It's just an arbitrary triangle.

00:03:27.150 --> 00:03:32.180
And so let's say this
coordinate right over here

00:03:32.180 --> 00:03:34.660
is-- I'll call this the x-axis.

00:03:34.660 --> 00:03:37.975
So, this is the x-axis,
the y-axis, and the z-axis.

00:03:37.975 --> 00:03:40.350
I know some of y'all are used
to swapping these two axes,

00:03:40.350 --> 00:03:41.730
but it doesn't
make a difference.

00:03:41.730 --> 00:03:45.470
So let's call this coordinate
right here a, 0, 0.

00:03:45.470 --> 00:03:47.840
So it's a along the x-axis.

00:03:47.840 --> 00:03:51.960
Let's call this
coordinate 0, b, 0.

00:03:51.960 --> 00:03:57.299
And let's call this
coordinate up here, 0, 0, c.

00:03:57.299 --> 00:03:58.840
And if you connect
the points, you're

00:03:58.840 --> 00:04:06.940
going to have a
triangle just like that.

00:04:06.940 --> 00:04:10.145
Now, the centroid of a triangle,
especially in three dimensions.

00:04:10.145 --> 00:04:12.620
The centroid of a
triangle is just

00:04:12.620 --> 00:04:18.829
going to be the average of the
coordinates of the vertices.

00:04:18.829 --> 00:04:22.170
Or the coordinate
of the centroid

00:04:22.170 --> 00:04:24.280
here is just going
to be the average

00:04:24.280 --> 00:04:25.914
of the coordinates
of the vertices.

00:04:25.914 --> 00:04:27.330
So this coordinate
right over here

00:04:27.330 --> 00:04:29.560
is going to be-- so
for the x-coordinate,

00:04:29.560 --> 00:04:31.830
we have 0 plus 0 plus a.

00:04:31.830 --> 00:04:33.330
So we have three coordinates.

00:04:33.330 --> 00:04:35.380
They add up to a, and
we have to divide by 3.

00:04:35.380 --> 00:04:37.550
So it's a over 3.

00:04:37.550 --> 00:04:41.160
The y-coordinate is
going to b plus 0 plus 0.

00:04:41.160 --> 00:04:43.090
They add up to b, but
we have three of them,

00:04:43.090 --> 00:04:45.144
so the average is b over 3.

00:04:45.144 --> 00:04:47.310
And then same thing-- we
do it for the z-coordinate.

00:04:47.310 --> 00:04:50.282
The average is going
to be c, is c over 3.

00:04:50.282 --> 00:04:51.990
And I'm not proving
it to you right here.

00:04:51.990 --> 00:04:53.900
You could verify
it for yourself.

00:04:53.900 --> 00:04:55.690
But it's going to
be the average,

00:04:55.690 --> 00:04:57.970
that if you were to figure
out what this line is,

00:04:57.970 --> 00:05:02.450
this line is, and this line is,
this centroid, or this center

00:05:02.450 --> 00:05:05.060
of mass of this triangle,
if it had some mass,

00:05:05.060 --> 00:05:07.660
is just the average
of these coordinates.

00:05:07.660 --> 00:05:10.480
Now, what we want to do
is use this information.

00:05:10.480 --> 00:05:12.530
Let's just use this
coordinate right here

00:05:12.530 --> 00:05:15.710
and then compare just
using the distance formula.

00:05:15.710 --> 00:05:23.120
Let's compare this distance up
here in orange to this distance

00:05:23.120 --> 00:05:26.180
down here in yellow.

00:05:26.180 --> 00:05:28.740
And remember, this
point right over here--

00:05:28.740 --> 00:05:33.490
this is the median of this
bottom side right over here.

00:05:33.490 --> 00:05:36.880
It's just going to be the
average of these two points.

00:05:36.880 --> 00:05:39.270
And so the x-coordinate--
0 plus a over 2

00:05:39.270 --> 00:05:41.080
is going to be a over 2.

00:05:41.080 --> 00:05:44.380
b plus 0 over 2 is
going to be b over 2.

00:05:44.380 --> 00:05:45.815
And then it has
no z-coordinates,

00:05:45.815 --> 00:05:47.430
so it's just going to be 0.

00:05:47.430 --> 00:05:49.294
0 plus 0 over 2 is 0.

00:05:49.294 --> 00:05:51.460
So we know the coordinates
for this point that point

00:05:51.460 --> 00:05:52.145
and that point.

00:05:52.145 --> 00:05:53.770
So we can calculate
the yellow distance

00:05:53.770 --> 00:05:56.800
and we can calculate
the orange distance.

00:05:56.800 --> 00:05:58.800
So let's calculate
the orange distance.

00:05:58.800 --> 00:06:06.870
So that is going to be equal
to the square root of--

00:06:06.870 --> 00:06:08.640
and we just take the
difference of each

00:06:08.640 --> 00:06:09.850
of these points squared.

00:06:09.850 --> 00:06:12.890
So it's a over 3
minus 0 squared.

00:06:12.890 --> 00:06:16.010
So that's going to
be a squared over 9,

00:06:16.010 --> 00:06:18.870
plus b over 3 minus 0 squared.

00:06:18.870 --> 00:06:21.770
So that's b squared over 9.

00:06:21.770 --> 00:06:28.413
Plus c over 3 minus c,
which is negative 2/3.

00:06:31.910 --> 00:06:33.680
And we want to square that.

00:06:33.680 --> 00:06:40.750
So we're going to have
positive 4 over 9c squared.

00:06:40.750 --> 00:06:45.860
Did I do that right? c over 3,
so 1/3 minus 1 is negative 2/3.

00:06:45.860 --> 00:06:48.090
So this is negative 2/3 c.

00:06:48.090 --> 00:06:48.940
You square it.

00:06:48.940 --> 00:06:51.590
You're going to
get 4/9 c squared.

00:06:51.590 --> 00:06:53.630
So that's the orange distance.

00:06:53.630 --> 00:06:56.370
Now, let's calculate--
and if we want to do it,

00:06:56.370 --> 00:06:58.214
we can express
this-- let me express

00:06:58.214 --> 00:06:59.630
it a little bit
simpler than this.

00:06:59.630 --> 00:07:04.210
This is the same thing as the
square root of a squared plus b

00:07:04.210 --> 00:07:08.850
squared plus 4c squared over
the square root of 9, which

00:07:08.850 --> 00:07:15.200
is just equal to 3.

00:07:15.200 --> 00:07:18.410
Now let's do the same thing
with the yellow distance.

00:07:18.410 --> 00:07:23.490
So it's going to be equal
to the square root of-- so

00:07:23.490 --> 00:07:26.440
if we have a over
2 minus a over 3.

00:07:26.440 --> 00:07:31.560
So 1/2 minus 1/3-- that's the
same thing as 3/6 minus 2/6,

00:07:31.560 --> 00:07:33.437
so it's 1/6 a.

00:07:33.437 --> 00:07:39.880
1/6 a squared is
a squared over 36.

00:07:39.880 --> 00:07:43.430
b over 2 minus b
over 3 is b over 6.

00:07:43.430 --> 00:07:44.590
You square it.

00:07:44.590 --> 00:07:48.370
You get plus b squared over 36.

00:07:48.370 --> 00:07:51.660
And then finally you have
0 minus c over 3 squared.

00:07:51.660 --> 00:07:55.510
That's going to be
c squared over 9.

00:07:55.510 --> 00:07:58.560
But just so we get a common
denominator, c squared over 9

00:07:58.560 --> 00:08:04.720
is the same thing as
plus 4c squared over 36.

00:08:04.720 --> 00:08:09.910
And we can rewrite this as the
square root of a squared plus b

00:08:09.910 --> 00:08:15.100
squared plus 4c squared over 6.

00:08:15.100 --> 00:08:16.760
So you can see,
this distance right

00:08:16.760 --> 00:08:19.880
here, if you multiply this
orange distance by 1/2,

00:08:19.880 --> 00:08:22.380
you're going to get-- so if you
multiply the orange distance

00:08:22.380 --> 00:08:28.290
by 1/2 or if you divide it by
2, you get the yellow distance.

00:08:28.290 --> 00:08:30.740
So this is always going to
be twice the distance as this

00:08:30.740 --> 00:08:33.159
because we did this in the
most general possible way.

00:08:33.159 --> 00:08:35.890
We assumed nothing
about this triangle.

00:08:35.890 --> 00:08:38.450
So remember that little
property that the centroid,

00:08:38.450 --> 00:08:41.470
the intersection of the
medians-- the intersection

00:08:41.470 --> 00:08:49.010
happens 2/3 away from the vertex
or 1/3 the length of the median

00:08:49.010 --> 00:08:52.600
away from the midpoint
of the opposite side.

00:08:52.600 --> 00:08:54.474
And we can use that
property in-- well,

00:08:54.474 --> 00:08:56.390
we'lll probably use it
in a bunch of problems.

00:08:56.390 --> 00:08:58.527
But anyway, hopefully you
found that interesting.


2003 AIME II problem 5

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:05.440
"A cylindrical log has
a diameter of 12 inches.

00:00:05.440 --> 00:00:09.010
A wedge is cut from the log
by making two planar cuts that

00:00:09.010 --> 00:00:11.630
go entirely through the log.

00:00:11.630 --> 00:00:15.100
The first is perpendicular
to the axis of the cylinder,

00:00:15.100 --> 00:00:19.150
and the plane of the second
cut forms a 45 degree angle

00:00:19.150 --> 00:00:20.850
with the plane of the first cut.

00:00:20.850 --> 00:00:22.750
The intersection
of these two planes

00:00:22.750 --> 00:00:26.940
has exactly one point
in common with the log.

00:00:26.940 --> 00:00:29.440
The number of cubic
inches in the wedge

00:00:29.440 --> 00:00:33.280
can be expressed as n pi,
where n is a positive integer.

00:00:33.280 --> 00:00:34.630
Find n."

00:00:34.630 --> 00:00:35.680
So let's think about it.

00:00:35.680 --> 00:00:38.040
So let's just draw what
they're describing.

00:00:38.040 --> 00:00:39.700
So we have a
cylindrical log that

00:00:39.700 --> 00:00:41.590
has a diameter of 12 inches.

00:00:41.590 --> 00:00:43.700
So let's draw that.

00:00:43.700 --> 00:00:45.080
So let me draw it like this.

00:00:45.080 --> 00:00:47.830
So that is a cross
section of the log.

00:00:47.830 --> 00:00:50.880
It has a diameter of 12 inches.

00:00:50.880 --> 00:00:54.600
So this is 12 inches.

00:00:54.600 --> 00:00:57.410
And it is a cylinder,
so it looks like this.

00:00:57.410 --> 00:01:00.130
That is our log.

00:01:00.130 --> 00:01:02.740
That is the log in question.

00:01:02.740 --> 00:01:06.020
A wedge is cut from the log
by making two planar cuts that

00:01:06.020 --> 00:01:08.200
go entirely through the log.

00:01:08.200 --> 00:01:14.330
The first cut is perpendicular
to the axis of the cylinder.

00:01:14.330 --> 00:01:16.740
So it would go-- it'd
really just cut it--

00:01:16.740 --> 00:01:18.160
it would just cut it straight.

00:01:18.160 --> 00:01:19.701
That's another way
to think about it.

00:01:19.701 --> 00:01:22.180
It's perpendicular to
the axis of the cylinder.

00:01:22.180 --> 00:01:24.470
The axis of the cylinder
goes through the cylinder,

00:01:24.470 --> 00:01:25.230
like that.

00:01:25.230 --> 00:01:25.820
I don't want to do
that; it'll make

00:01:25.820 --> 00:01:27.620
the diagram a little confusing.

00:01:27.620 --> 00:01:31.130
If the log was transparent,
our cut would look like this.

00:01:31.130 --> 00:01:33.840
This first cut would just go
through the log like this.

00:01:33.840 --> 00:01:37.500
It would really just form
a cross section of the log.

00:01:37.500 --> 00:01:38.500
So that's the first cut.

00:01:38.500 --> 00:01:42.725
The first is perpendicular
to the axis of the cylinder.

00:01:42.725 --> 00:01:47.130
And the plane of the second
cut forms a 45 degree angle

00:01:47.130 --> 00:01:49.920
with the plane of the first cut.

00:01:49.920 --> 00:01:52.850
So that thing is going to
come in at a 45 degree angle.

00:01:52.850 --> 00:01:57.260
So it's going to cut into
our log something like that.

00:01:57.260 --> 00:02:01.400
Something like that,
right over there.

00:02:01.400 --> 00:02:02.900
The intersection
of these two planes

00:02:02.900 --> 00:02:05.620
has exactly one point
in common with the log.

00:02:05.620 --> 00:02:09.060
That would be this
point, right over here.

00:02:09.060 --> 00:02:11.190
So we need to find the
number of cubic inches

00:02:11.190 --> 00:02:13.460
in the wedge, expressed
as n pi, and then we

00:02:13.460 --> 00:02:14.390
have to figure out n.

00:02:14.390 --> 00:02:18.080
So let's draw this wedge, this
wedge that we've cut out here.

00:02:18.080 --> 00:02:20.080
So this right here is our wedge.

00:02:20.080 --> 00:02:22.010
Let me take it out and
kind of flip it around.

00:02:22.010 --> 00:02:23.750
So the base of--
I'll make the base

00:02:23.750 --> 00:02:26.000
of the wedge the
perpendicular cut.

00:02:26.000 --> 00:02:27.310
This cut right over here.

00:02:27.310 --> 00:02:30.950
So that is the
base of the wedge.

00:02:30.950 --> 00:02:34.020
That is our base.

00:02:34.020 --> 00:02:37.430
And then you can do the top
of our cut to be the magenta,

00:02:37.430 --> 00:02:39.230
the 45 degree angle cut.

00:02:39.230 --> 00:02:41.200
So maybe I'll draw it like this.

00:02:41.200 --> 00:02:45.410
So it would look
something like this.

00:02:45.410 --> 00:02:47.780
I'm trying my best to draw it.

00:02:47.780 --> 00:02:49.970
That's really the
hard part, here.

00:02:49.970 --> 00:02:55.380
So let me draw the
45 degree angle cut.

00:02:55.380 --> 00:02:58.930
It's going to look
something like that, where

00:02:58.930 --> 00:03:00.940
this angle right over
here-- so if I were

00:03:00.940 --> 00:03:03.819
to go-- the diameter of this
top thing, it's actually,

00:03:03.819 --> 00:03:05.360
it's not going to
be a normal circle.

00:03:05.360 --> 00:03:07.010
It's going to be
more elliptical.

00:03:07.010 --> 00:03:09.700
But the diameter
of this top thing

00:03:09.700 --> 00:03:12.270
versus the diameter
of the base, which

00:03:12.270 --> 00:03:14.170
is the diameter of a
circle, this right here

00:03:14.170 --> 00:03:17.720
is going to be a
45 degree angle.

00:03:17.720 --> 00:03:20.237
Now when I first
looked at this problem,

00:03:20.237 --> 00:03:21.820
there's all sorts
of temptations here.

00:03:21.820 --> 00:03:23.350
Maybe you use some calculus.

00:03:23.350 --> 00:03:27.340
Maybe you use some-- you rotate
something around some axis

00:03:27.340 --> 00:03:28.380
to find the volume.

00:03:28.380 --> 00:03:31.130
Maybe you can take some
type of average, here.

00:03:31.130 --> 00:03:33.299
And you probably could
do something like that.

00:03:33.299 --> 00:03:34.840
But the easiest
thing here-- and this

00:03:34.840 --> 00:03:36.240
is kind of-- whenever
you see these kind

00:03:36.240 --> 00:03:37.790
of competition-type
math problems--

00:03:37.790 --> 00:03:42.000
and this comes from
the 2003 AIM exam--

00:03:42.000 --> 00:03:44.710
is that there should be
a quick way to do it.

00:03:44.710 --> 00:03:46.180
And for this exam,
particular, you

00:03:46.180 --> 00:03:47.972
shouldn't have to use
any type of calculus.

00:03:47.972 --> 00:03:49.805
And so, if you find
yourself doing something

00:03:49.805 --> 00:03:51.390
tedious and hairy,
you're probably not

00:03:51.390 --> 00:03:54.230
seeing the easiest way
to do this problem.

00:03:54.230 --> 00:03:57.160
And this problem is actually
ridiculously easy to do,

00:03:57.160 --> 00:03:58.770
if you just see the trick here.

00:03:58.770 --> 00:04:01.130
And the trick here is,
instead of just directly

00:04:01.130 --> 00:04:04.890
trying to solve for the volume
of this figure, right here,

00:04:04.890 --> 00:04:08.540
take another one of these guys,
and flip it over, and put it

00:04:08.540 --> 00:04:09.510
on top of him.

00:04:09.510 --> 00:04:11.010
So let's say, if
you did that, you

00:04:11.010 --> 00:04:13.260
would have another thing on
top of it, just like that.

00:04:13.260 --> 00:04:15.262
So if I just took
two of these wedges

00:04:15.262 --> 00:04:17.720
and stacked them on top of each
other-- flipped one of them

00:04:17.720 --> 00:04:23.590
and stacked them on top of each
other, it would look like that.

00:04:23.590 --> 00:04:25.120
This is another
wedge right here.

00:04:25.120 --> 00:04:28.240
So I took their angled
faces and put them

00:04:28.240 --> 00:04:30.300
right on top of each other.

00:04:30.300 --> 00:04:33.460
And so if you took two wedges
together, flip one of them,

00:04:33.460 --> 00:04:35.642
and put it on top of the
other, what do you get?

00:04:35.642 --> 00:04:38.100
So the equivalent-- I could
draw the green wedge right over

00:04:38.100 --> 00:04:38.900
here.

00:04:38.900 --> 00:04:41.250
The green wedge
would look like this,

00:04:41.250 --> 00:04:43.440
where its base looks like this.

00:04:43.440 --> 00:04:46.450
So what would two wedges put in
this configuration look like?

00:04:46.450 --> 00:04:47.850
Well, it's just a cylinder, now.

00:04:47.850 --> 00:04:49.475
And it now is a cylinder.

00:04:52.580 --> 00:04:55.510
So this is a cylinder
with a diameter of 12.

00:04:55.510 --> 00:04:57.100
So that diameter is 12.

00:04:57.100 --> 00:05:00.060
But in order to figure out
the volume of a cylinder,

00:05:00.060 --> 00:05:03.214
you still need to figure out
the height of the cylinder.

00:05:03.214 --> 00:05:05.630
You still have to figure out
this length, right over here.

00:05:05.630 --> 00:05:08.125
You still have to figure out
what the height is equal to.

00:05:08.125 --> 00:05:10.500
You have to figure out what
this length, right over here,

00:05:10.500 --> 00:05:11.540
is equal to.

00:05:11.540 --> 00:05:13.545
And that's where the
45 degrees helps us.

00:05:13.545 --> 00:05:15.350
Well, the 45 degrees
already helped us,

00:05:15.350 --> 00:05:18.270
because if you flipped the
thing over and put it on top,

00:05:18.270 --> 00:05:20.751
it forms a nice cylinder for us.

00:05:20.751 --> 00:05:22.250
If it was another
angle, it wouldn't

00:05:22.250 --> 00:05:24.350
have been a nice, clean
cylinder, like this.

00:05:24.350 --> 00:05:27.980
But the 45 degrees also
tells us what this height is.

00:05:27.980 --> 00:05:30.120
And let's just think about
it, here, for a second.

00:05:30.120 --> 00:05:32.203
This triangle that I had
drawn, in the beginning--

00:05:32.203 --> 00:05:35.179
let me draw this in--
well, I already used blue.

00:05:35.179 --> 00:05:36.220
Let me draw it in yellow.

00:05:36.220 --> 00:05:38.990
So if I were to
take the diameter

00:05:38.990 --> 00:05:43.912
of the angled surface-- and
once again, it's not a circle,

00:05:43.912 --> 00:05:45.620
but it's kind of a
stretched-out circle--

00:05:45.620 --> 00:05:48.360
and I took the
diameter of the base,

00:05:48.360 --> 00:05:50.940
they form a 45 degree angle.

00:05:50.940 --> 00:05:52.870
So this is 45 degrees.

00:05:52.870 --> 00:05:56.310
This right here is also
going to be 45 degrees.

00:05:56.310 --> 00:05:58.350
This length, over
here, we know--

00:05:58.350 --> 00:06:02.690
we know this length, right
over here, is going to be 12.

00:06:02.690 --> 00:06:06.130
Now, this right here
is a 45-45-90 triangle.

00:06:06.130 --> 00:06:08.390
Let me draw it like this.

00:06:08.390 --> 00:06:11.200
I could draw it like this.

00:06:11.200 --> 00:06:15.367
This right here is
45-45-90 triangle.

00:06:15.367 --> 00:06:17.200
And you might say, how
did I know that's 45?

00:06:17.200 --> 00:06:19.780
Well, this is going to be a
right angle, right over here.

00:06:19.780 --> 00:06:22.120
And the sum of the angles
have to add up to 180.

00:06:22.120 --> 00:06:24.170
And then, if you have
45 and 90 already, then

00:06:24.170 --> 00:06:26.570
this one has to be 45 degrees.

00:06:26.570 --> 00:06:31.150
And in 45-45-90 triangle, this
side is equal to that side.

00:06:31.150 --> 00:06:32.816
It is an isosceles triangle.

00:06:32.816 --> 00:06:34.190
The two base angles
are the same,

00:06:34.190 --> 00:06:36.170
so the two sides are
going to be the same.

00:06:36.170 --> 00:06:38.890
So if this side, right
over here, is 12,

00:06:38.890 --> 00:06:41.690
then this side, right
over here-- this side,

00:06:41.690 --> 00:06:43.320
right over here is 12.

00:06:43.320 --> 00:06:46.020
So the height is going
to be equal to 12.

00:06:46.020 --> 00:06:48.650
So let's figure out the
volume of this cylinder that's

00:06:48.650 --> 00:06:50.390
essentially two
wedges, and then we

00:06:50.390 --> 00:06:54.590
can take half of that to
find the volume of one wedge.

00:06:54.590 --> 00:06:56.550
And to find the
volume of a cylinder,

00:06:56.550 --> 00:07:00.990
you just have to find the area
of the top of the cylinder.

00:07:00.990 --> 00:07:04.450
And to find that, it's going to
be pi times the radius squared.

00:07:04.450 --> 00:07:06.770
The radius here is 6, 1/2 of 12.

00:07:06.770 --> 00:07:09.090
So the area is pi r squared.

00:07:09.090 --> 00:07:12.290
So it's 36 pi--
that's that area--

00:07:12.290 --> 00:07:17.720
times the height-- times 12.

00:07:17.720 --> 00:07:19.620
And so the whole
volume, our volume

00:07:19.620 --> 00:07:21.810
is going to be equal
to-- what's that?

00:07:21.810 --> 00:07:23.770
360 plus 70.

00:07:23.770 --> 00:07:24.960
Let me just multiply it out.

00:07:24.960 --> 00:07:26.626
I don't want to make
a careless mistake.

00:07:26.626 --> 00:07:27.660
36 times 12.

00:07:27.660 --> 00:07:30.470
36 times 2 is 72.

00:07:30.470 --> 00:07:33.070
1 times 36 is 36.

00:07:33.070 --> 00:07:35.590
2, 13, 4.

00:07:35.590 --> 00:07:39.660
So it's 432 pi.

00:07:39.660 --> 00:07:41.110
Now we have to be very careful.

00:07:41.110 --> 00:07:43.710
This is the volume
of two wedges.

00:07:43.710 --> 00:07:46.300
So this is the volume of
two wedges, I could call it.

00:07:46.300 --> 00:07:51.656
So the volume of one wedge,
is going to be 1/2 of this.

00:07:51.656 --> 00:07:53.500
Let me do this in
a different color.

00:07:53.500 --> 00:07:59.780
The volume of one wedge is going
to be 1/2 of this, or 216 pi.

00:07:59.780 --> 00:08:02.350
And so if we want to
find n, because they say,

00:08:02.350 --> 00:08:04.170
the number of cubic
inches in the wedge

00:08:04.170 --> 00:08:05.870
can be expressed as n pi.

00:08:05.870 --> 00:08:08.380
It's 216 pi, where n
is a positive integer.

00:08:08.380 --> 00:08:09.000
Find n.

00:08:09.000 --> 00:08:12.850
Well, we just figured
that out. n is 216.


Sum of factors 2

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.310
In the last video, we
kind of reasoned our way

00:00:03.310 --> 00:00:08.234
towards finding the sum
of the factors of 27,000.

00:00:08.234 --> 00:00:10.150
What I want to do in
this video is really just

00:00:10.150 --> 00:00:11.960
revisit what we did
in the last video,

00:00:11.960 --> 00:00:15.670
and see if we can come up with a
more generalized way of finding

00:00:15.670 --> 00:00:18.161
the sum of the
factors of a number.

00:00:18.161 --> 00:00:20.035
So in the last video,
we started with 27,000.

00:00:20.035 --> 00:00:22.500
And we figured out that
its prime factorization

00:00:22.500 --> 00:00:27.160
is 3 to the third, times
2 to the third, times 5

00:00:27.160 --> 00:00:27.730
to the third.

00:00:27.730 --> 00:00:28.490
Or we could have
said 2 the third--

00:00:28.490 --> 00:00:30.490
we could have switched
the orders here.

00:00:30.490 --> 00:00:33.700
And we used that information
to say, well, OK, if this

00:00:33.700 --> 00:00:37.920
is the prime factorization of
27,000, than any factor-- so,

00:00:37.920 --> 00:00:40.790
let's say that this is
some factor of 27,000--

00:00:40.790 --> 00:00:45.187
it has to be the product
of powers of 3, 2, and 5.

00:00:45.187 --> 00:00:46.270
So let me make that clear.

00:00:46.270 --> 00:00:50.660
It has to be 3 to something
times 2 to something else--

00:00:50.660 --> 00:00:55.040
they could be the same number--
times 5 to something else.

00:00:55.040 --> 00:01:01.200
And each of these x, y's, and
z's have to be between 0 and 3,

00:01:01.200 --> 00:01:03.760
and they could be
equal to 0 or 3.

00:01:03.760 --> 00:01:06.610
And actually, make it be clear--
they have to be integers.

00:01:06.610 --> 00:01:10.280
They have to be integers,
where these are integers.

00:01:10.280 --> 00:01:11.560
Integers.

00:01:11.560 --> 00:01:14.410
This notation here implies that
you could be any number there,

00:01:14.410 --> 00:01:17.430
but they have to be
integers in between 0 and 3,

00:01:17.430 --> 00:01:19.050
including 0 and 3.

00:01:19.050 --> 00:01:21.870
And just to be clear, we can
just generate arbitrary factors

00:01:21.870 --> 00:01:22.030
here.

00:01:22.030 --> 00:01:23.446
We could say, OK,
let's pick zero.

00:01:23.446 --> 00:01:28.970
It could be 3 to the 0, times
2 to the second power, times 5

00:01:28.970 --> 00:01:29.590
to the 1.

00:01:29.590 --> 00:01:32.500
This will be a factor of 27,000.

00:01:32.500 --> 00:01:35.700
You could figure out this
is 1, times 4, times 5.

00:01:35.700 --> 00:01:37.190
This is equal to 20.

00:01:37.190 --> 00:01:38.700
And you can just
look-- you can just

00:01:38.700 --> 00:01:40.579
keep taking different
values of x, y,

00:01:40.579 --> 00:01:41.870
and z to get all of the factor.

00:01:41.870 --> 00:01:43.495
So essentially, you
have to through all

00:01:43.495 --> 00:01:45.792
of the different combinations
here and add them up.

00:01:45.792 --> 00:01:47.500
And that's what we
did in the last video.

00:01:47.500 --> 00:01:49.041
And to be able to
think through that,

00:01:49.041 --> 00:01:51.060
I drew a little bit of a grid.

00:01:51.060 --> 00:01:52.650
I started with powers of 3.

00:01:52.650 --> 00:01:55.070
So 3 to the 0-- let
me write it here.

00:01:55.070 --> 00:01:58.210
So you start with 3 to
the 0, 3 to the 1, 3

00:01:58.210 --> 00:02:02.130
squared, 3 to the third is 27.

00:02:02.130 --> 00:02:05.240
And then I had powers
of 2-- 2 to the 0,

00:02:05.240 --> 00:02:10.080
2 to the first, 2
squared, 2 squared is 4.

00:02:10.080 --> 00:02:12.320
And then 2 the third is 8.

00:02:12.320 --> 00:02:15.770
And here, I was just
looking for the combinations

00:02:15.770 --> 00:02:18.550
where we're looking at
the powers of 3 and 2

00:02:18.550 --> 00:02:20.810
and we're multiplying
it times 5 to the 0.

00:02:20.810 --> 00:02:23.250
So, we're not even
looking at having

00:02:23.250 --> 00:02:27.592
a 5 as a factor yet of
all of these combinations.

00:02:27.592 --> 00:02:29.800
And what we did is we just
took each of these numbers

00:02:29.800 --> 00:02:32.260
and we multiplied
them by each other.

00:02:32.260 --> 00:02:36.070
So these would just be
a 1, a 3, a 9, and a 27.

00:02:36.070 --> 00:02:39.940
I just took 1 times 1
is 1, 3 times 1 is 3.

00:02:39.940 --> 00:02:42.180
And then you could go
down this row-- 1 times 2

00:02:42.180 --> 00:02:45.190
is 2, 3 times 2 is 6.

00:02:45.190 --> 00:02:46.510
9 times 2 is 18.

00:02:46.510 --> 00:02:49.200
You could just keep
going-- 27 times 2 is 54.

00:02:49.200 --> 00:02:52.000
And you could keep
figuring out all of these.

00:02:52.000 --> 00:02:54.050
But the shortcut that
we did in the last video

00:02:54.050 --> 00:02:56.660
is we said, look, you don't
have to figure out all of those.

00:02:56.660 --> 00:02:58.270
You could just
take this sum over

00:02:58.270 --> 00:03:00.140
here-- so whatever this is.

00:03:00.140 --> 00:03:04.890
So 1 plus 3, plus 9, plus 27.

00:03:04.890 --> 00:03:07.620
We figured out in the
last video it was 40.

00:03:07.620 --> 00:03:09.280
This is the sum for
this first column.

00:03:09.280 --> 00:03:12.300
The second sum is just going
to be 2 times this number.

00:03:12.300 --> 00:03:13.490
So let me make it clear.

00:03:13.490 --> 00:03:16.060
This is the same
thing as 1 times this.

00:03:16.060 --> 00:03:21.770
The second column is going
to be 2 times this thing.

00:03:21.770 --> 00:03:24.440
Because each of these terms
are just being multiplied by 2.

00:03:24.440 --> 00:03:30.960
So it's 2 times 1, plus
3, plus 9, plus 27.

00:03:30.960 --> 00:03:34.680
And then the third column
is going to be 4 times.

00:03:34.680 --> 00:03:37.510
It's going to be 4 times
this business over here.

00:03:37.510 --> 00:03:41.190
So 4 times-- let me
scroll over-- 4 times

00:03:41.190 --> 00:03:45.020
1, plus 3, plus 9, plus 27.

00:03:45.020 --> 00:03:47.890
And then the fourth
column will be 8 times,

00:03:47.890 --> 00:03:51.480
and then they'll be plus
8 times all of that stuff.

00:03:51.480 --> 00:03:54.530
1 plus 3, plus 9, plus 27.

00:03:54.530 --> 00:03:56.720
In the last video, I
added them up and got 40,

00:03:56.720 --> 00:03:59.210
but I'm keeping them
this way on purpose.

00:03:59.210 --> 00:04:02.100
Because this thing right here,
what I just wrote-- and this

00:04:02.100 --> 00:04:07.360
is essentially the sums
of all of the combinations

00:04:07.360 --> 00:04:11.310
where our x's and y's are
integers between 0 and 3,

00:04:11.310 --> 00:04:12.700
including 0 and 3.

00:04:12.700 --> 00:04:16.060
And z is 0, because we're not
multiplying any of these by 5.

00:04:16.060 --> 00:04:18.695
So when we ignore
the 5, the sum here--

00:04:18.695 --> 00:04:20.820
and this is exactly what
we did in the last video--

00:04:20.820 --> 00:04:26.040
this is exactly
equal to 1 plus 3,

00:04:26.040 --> 00:04:35.630
plus 9, plus 27, times 1
plus 2, plus 4, plus 8.

00:04:35.630 --> 00:04:39.280
If you just take
this term right here

00:04:39.280 --> 00:04:44.620
and distribute it onto
each of these terms

00:04:44.620 --> 00:04:46.640
you will get this
expression up here.

00:04:46.640 --> 00:04:49.940
So to find all of the
combinations of the powers of 2

00:04:49.940 --> 00:04:54.660
and 3 with not
multiplying things by 5,

00:04:54.660 --> 00:04:56.820
you, literally, just have
to take the powers of 3

00:04:56.820 --> 00:04:58.640
up to the third
power, because that's

00:04:58.640 --> 00:05:01.020
how many powers-- that's
how high we get when

00:05:01.020 --> 00:05:05.050
we do the prime factorization
of 27,000-- and sum them up.

00:05:05.050 --> 00:05:07.470
And then multiply that
times the power of 2

00:05:07.470 --> 00:05:08.747
up to the third power.

00:05:08.747 --> 00:05:11.080
And then, what we did in the
last video, we figured out.

00:05:11.080 --> 00:05:13.320
OK, this value
right here-- this is

00:05:13.320 --> 00:05:16.612
when we're only dealing
with 5 to the 0 power.

00:05:16.612 --> 00:05:18.070
If we want all of
the combinations,

00:05:18.070 --> 00:05:20.560
we would want this
grid multiplied by 5

00:05:20.560 --> 00:05:22.944
to the 0, which is essentially
what this grid already is.

00:05:22.944 --> 00:05:24.360
And then we would
want to add that

00:05:24.360 --> 00:05:27.080
to that grid multiplied
by 5 to the first.

00:05:27.080 --> 00:05:32.360
And then, add that to this
grid multiplied by 5 squared.

00:05:32.360 --> 00:05:34.050
And then add that
to this whole grid

00:05:34.050 --> 00:05:36.220
multiplied by 5 to the third.

00:05:36.220 --> 00:05:39.240
And so this whole grid--
the sum of this whole grid--

00:05:39.240 --> 00:05:40.390
is this right over here.

00:05:40.390 --> 00:05:41.910
Let's just call
this, I don't know.

00:05:41.910 --> 00:05:44.570
For simplicity, let's
just call this A.

00:05:44.570 --> 00:05:49.800
So if you really want the sum
of all of the factors of 27,000,

00:05:49.800 --> 00:05:54.360
you would take A and multiply
it by 5 to the 0, which is 1.

00:05:54.360 --> 00:05:57.720
And then add that
to A multiplied by 5

00:05:57.720 --> 00:06:01.540
squared-- or sorry, multiplied
by 5 to the first, which is 5.

00:06:01.540 --> 00:06:04.200
And then add that
to A multiplied

00:06:04.200 --> 00:06:08.590
by 5 squared, which is 25.

00:06:08.590 --> 00:06:11.510
And then add that
to A multiplied by 5

00:06:11.510 --> 00:06:13.417
to the third, which is 125.

00:06:13.417 --> 00:06:15.250
And of course, you can
factor out an A here.

00:06:15.250 --> 00:06:16.291
So this is our total sum.

00:06:16.291 --> 00:06:19.175
This would be the sum of
all the factors of 27,000.

00:06:19.175 --> 00:06:21.180
But you could factor
and A out here.

00:06:21.180 --> 00:06:28.530
And so our sum would be A times
1, plus 5, plus 25, plus 125.

00:06:28.530 --> 00:06:29.840
And we know what A is.

00:06:29.840 --> 00:06:31.210
A is this thing over here.

00:06:31.210 --> 00:06:34.140
So the sum of all of the
factors of 27,000-- actually,

00:06:34.140 --> 00:06:36.370
let me keep this over
here-- is equal to--

00:06:36.370 --> 00:06:40.160
and I'll color code it--
it's equal to-- this is A.

00:06:40.160 --> 00:06:44.790
So A is 1 plus 3,
plus 9, plus 27--

00:06:44.790 --> 00:06:46.300
it's not the right
shade of green,

00:06:46.300 --> 00:06:50.420
but hopefully, you get the
point-- times 1 plus 2,

00:06:50.420 --> 00:07:01.900
plus 4, plus 8, times 1
plus 5, plus 25, plus 125.

00:07:01.900 --> 00:07:04.320
So this, essentially,
is giving you

00:07:04.320 --> 00:07:08.060
a formula for finding the sum
of the factors of a number.

00:07:08.060 --> 00:07:10.572
You do the prime
factorization of the number.

00:07:10.572 --> 00:07:12.030
And I'm not doing
in general terms,

00:07:12.030 --> 00:07:13.520
I'm doing with a
particular case.

00:07:13.520 --> 00:07:14.936
And I'm doing that
because I think

00:07:14.936 --> 00:07:17.510
it makes it a little bit
clearer why this works.

00:07:17.510 --> 00:07:20.230
But our prime
factorization of 27,000

00:07:20.230 --> 00:07:22.890
is 3 to the third, times
2 to the third, times 5

00:07:22.890 --> 00:07:23.616
to the third.

00:07:23.616 --> 00:07:24.990
If we really want
to do the sum--

00:07:24.990 --> 00:07:26.650
and this is exactly what
we did in the last video,

00:07:26.650 --> 00:07:28.983
I'm just expressing it slightly
different here-- we just

00:07:28.983 --> 00:07:32.700
take all of the powers of 3 up
to 3 to the third, add them up.

00:07:32.700 --> 00:07:35.720
All the powers of 2 up to 2
to the third, add them up.

00:07:35.720 --> 00:07:39.080
All the powers of 5 up to 5
to the third, add them up.

00:07:39.080 --> 00:07:40.830
And then, take the product.

00:07:40.830 --> 00:07:42.140
And we can do that right now.

00:07:42.140 --> 00:07:47.810
So this is going to be-- so
this first term over here is 40,

00:07:47.810 --> 00:07:52.770
this term right here, 40
times-- this is 4 plus 8 is 12,

00:07:52.770 --> 00:07:55.070
plus 3 is 15.

00:07:55.070 --> 00:07:55.570
Right?

00:07:55.570 --> 00:07:56.070
3, yup.

00:07:56.070 --> 00:07:56.910
15.

00:07:56.910 --> 00:08:01.300
And then times--
25 plus 125 is 150.

00:08:01.300 --> 00:08:05.160
So this is 156.

00:08:05.160 --> 00:08:06.780
And if you take
the product-- here,

00:08:06.780 --> 00:08:09.770
actually, let's just do
it just to prove the point

00:08:09.770 --> 00:08:11.600
that this actually does work.

00:08:11.600 --> 00:08:15.960
So 40 times 15--
4 times 15 is 60--

00:08:15.960 --> 00:08:18.230
so this is going to be 600.

00:08:18.230 --> 00:08:20.790
So this part right here is
going to be equal to 600.

00:08:20.790 --> 00:08:24.094
So it's going to boil
down to 600 times 156.

00:08:24.094 --> 00:08:25.010
So let's just do that.

00:08:25.010 --> 00:08:28.160
156-- let's just
multiply it by 6,

00:08:28.160 --> 00:08:30.380
and then I'll add two
0's to it at the end.

00:08:30.380 --> 00:08:33.030
So 6 times 6 is 36.

00:08:33.030 --> 00:08:37.270
6 times 5 is 30, plus 3 is 33.

00:08:37.270 --> 00:08:39.780
6 times 1 is 6, plus 3 is 9.

00:08:39.780 --> 00:08:42.070
And of course, if we want
to multiply it by 600,

00:08:42.070 --> 00:08:45.250
we just add two 0's here, and
we get the exact same answer

00:08:45.250 --> 00:08:47.781
that we got in the last video.

00:08:47.781 --> 00:08:49.280
So hopefully, you
found that useful.

00:08:49.280 --> 00:08:50.940
I haven't proven it
in general terms,

00:08:50.940 --> 00:08:52.314
but you can--
maybe I'll do that,

00:08:52.314 --> 00:08:54.240
if I feel like doing
a video on that.

00:08:54.240 --> 00:08:57.660
But you see a general
method for finding

00:08:57.660 --> 00:08:59.220
the sum of the
factors of a number.

00:08:59.220 --> 00:09:01.030
Actually, let me
just write that.

00:09:01.030 --> 00:09:03.589
Let's just do another
problem just real fast.

00:09:03.589 --> 00:09:05.380
Let's say I had the
number-- I don't know--

00:09:05.380 --> 00:09:11.610
let's say we had to find the
sum of the factors of 300.

00:09:11.610 --> 00:09:18.400
So 300 is 3 times 100,
which is 10 times 10.

00:09:18.400 --> 00:09:22.780
And that is 2 times
5, times 2, times 5.

00:09:22.780 --> 00:09:30.390
So 300 is equal to 3 times
2 squared, times 5 squared.

00:09:30.390 --> 00:09:32.750
That's the prime
factorization of 300.

00:09:32.750 --> 00:09:35.570
So if we wanted to find the
sum of all of its factors,

00:09:35.570 --> 00:09:40.290
it would be 3 to
the 0, which is 1,

00:09:40.290 --> 00:09:42.630
plus 3 to the 1, which is three.

00:09:42.630 --> 00:09:44.430
We won't go to any
higher powers of 3

00:09:44.430 --> 00:09:46.013
here, because that's
the highest power

00:09:46.013 --> 00:09:47.590
we have in the
prime factorization.

00:09:47.590 --> 00:09:49.850
So that's dealing with the 3's.

00:09:49.850 --> 00:09:54.610
Times 2 to the 0,
which is 1, plus 2

00:09:54.610 --> 00:09:57.980
to the 1, which is 2,
plus 2 squared-- that's

00:09:57.980 --> 00:10:00.170
how high we can get, which is 4.

00:10:00.170 --> 00:10:06.140
And then, times-- you just
do the 5-- times 5 to the 0,

00:10:06.140 --> 00:10:08.410
which is 1, plus 5 to
the first power, which

00:10:08.410 --> 00:10:10.504
is 5, plus 5
squared, which is 25.

00:10:10.504 --> 00:10:12.420
Can't go any higher than
that, because we only

00:10:12.420 --> 00:10:15.510
go up to two 5's up here
at 5 to the second power.

00:10:15.510 --> 00:10:26.740
And so this is going to
be 4 times 7, times 31.

00:10:26.740 --> 00:10:28.710
So this is going
to be 28 times 31.

00:10:32.050 --> 00:10:34.180
1 times 28 is 28.

00:10:34.180 --> 00:10:34.930
Put a 0 down here.

00:10:34.930 --> 00:10:38.370
3 times 8 is 24.

00:10:38.370 --> 00:10:46.400
3 times 2 is 6, plus 2 is 8,
and then you get 8, 6, and 8.

00:10:46.400 --> 00:10:51.030
So the sum of all the
factors of 300 is 868.

00:10:51.030 --> 00:10:53.610
Hopefully, you found
that interesting.


Sum of factors of 27000

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:05.280
Calculate the sum of all
positive divisors of 27,000.

00:00:05.280 --> 00:00:07.100
The easiest thing that
I can think of doing

00:00:07.100 --> 00:00:10.237
is first take the prime
factorization of 27,000,

00:00:10.237 --> 00:00:12.570
and then that will help us
kind of structure our thought

00:00:12.570 --> 00:00:16.370
of what all of the
different divisors of 27,000

00:00:16.370 --> 00:00:18.020
would have to look like.

00:00:18.020 --> 00:00:26.140
So 27,000 is the same thing
as 27 times 1,000, which

00:00:26.140 --> 00:00:30.470
is the same thing as 3 to the
third times 10 to the third,

00:00:30.470 --> 00:00:33.590
and 10 is, of course, the
same thing as 2 times 5.

00:00:33.590 --> 00:00:37.690
So this is the same thing
as 2 times 5 to the third,

00:00:37.690 --> 00:00:41.920
or it's the same thing as 2 to
the third times 5 to the third.

00:00:41.920 --> 00:00:48.560
So 27,000 is equal to
2 to the third times 3

00:00:48.560 --> 00:00:52.660
to the third times
5 to the third.

00:00:52.660 --> 00:00:56.580
So any divisor of 27,000 is
going to have to be made up

00:00:56.580 --> 00:01:02.590
of the product of up to three
2's, up to three 3's, and up

00:01:02.590 --> 00:01:03.626
to three 5's.

00:01:03.626 --> 00:01:05.459
So let's try to look
at all the combinations

00:01:05.459 --> 00:01:08.000
and think of a fast
way of summing them.

00:01:08.000 --> 00:01:10.170
So let's just say it
has no fives in it.

00:01:10.170 --> 00:01:12.010
It has no fives in a divisor.

00:01:12.010 --> 00:01:16.710
So if it has no fives, then
it could have up to three

00:01:16.710 --> 00:01:19.071
2's, so let's say
it has zero 2's.

00:01:19.071 --> 00:01:20.820
So I'm just going to
take the powers of 2,

00:01:20.820 --> 00:01:22.900
so if it has zero 2's,
then we'll put a 1

00:01:22.900 --> 00:01:25.960
here, if it has two 2's, it
has to be divisible by 4.

00:01:25.960 --> 00:01:28.860
If it has three 2's, it's
going to be divisible by 8.

00:01:28.860 --> 00:01:32.030
When I say three 2's, I
mean 2 times 2 times 2.

00:01:32.030 --> 00:01:35.590
Now, let's do it with the 3's.

00:01:35.590 --> 00:01:37.810
If you have, oh wait,
I forgot a power.

00:01:37.810 --> 00:01:40.660
If you have zero 2's, that means
it's just divisible by 1 from

00:01:40.660 --> 00:01:41.630
looking at the 2's.

00:01:41.630 --> 00:01:44.830
If you have one 2, it's
divisible by just 2.

00:01:44.830 --> 00:01:47.460
If you have two 2's,
you're divisible by 4.

00:01:47.460 --> 00:01:49.050
And if you have
three 2's, and when

00:01:49.050 --> 00:01:51.060
I mean that I'm saying
2 times 2 times 2,

00:01:51.060 --> 00:01:53.630
that means you're
divisible by 8.

00:01:53.630 --> 00:01:55.716
Let's do the same thing with 3.

00:01:55.716 --> 00:01:58.340
From the point of view of the 3,
if you have no 3's, that means

00:01:58.340 --> 00:02:00.460
at least you're divisible by 1.

00:02:00.460 --> 00:02:03.650
If you have one 3, that
means you're divisible by 3.

00:02:03.650 --> 00:02:07.800
Two 3's, or 3 times 3 means
you're divisible by 9.

00:02:07.800 --> 00:02:11.330
If you have three 3's, it
means you're divisible by 27.

00:02:11.330 --> 00:02:13.780
So let's look at all of
the possible combinations.

00:02:13.780 --> 00:02:16.440
And for this grid that I'm
going to generate right here,

00:02:16.440 --> 00:02:18.760
we assume that you're
not divisible by 5,

00:02:18.760 --> 00:02:22.900
or you're only divisible
by 5 to the zero power.

00:02:22.900 --> 00:02:24.970
So what are all the
possible numbers here?

00:02:24.970 --> 00:02:26.720
Well, you have 1 times 1 is 1.

00:02:26.720 --> 00:02:28.020
That's divisible by 1 and 1.

00:02:28.020 --> 00:02:33.190
You have 1 times 3, which
is 3, 1 times 9 which is 9,

00:02:33.190 --> 00:02:35.480
1 times 27 which is 27.

00:02:35.480 --> 00:02:36.950
So these are all
the numbers that

00:02:36.950 --> 00:02:40.760
are divisible by that have up
to three 3's in them, from zero

00:02:40.760 --> 00:02:43.265
to three 3's in them, and
they have no twos in them.

00:02:43.265 --> 00:02:44.640
If you throw
another two in here,

00:02:44.640 --> 00:02:46.240
you're essentially going to
multiply all of these numbers

00:02:46.240 --> 00:02:47.170
by two.

00:02:47.170 --> 00:02:48.655
If you throw
another two in here,

00:02:48.655 --> 00:02:50.780
you're going to multiply
all of these numbers by 2.

00:02:50.780 --> 00:02:52.090
Now, before I do
this, because I want

00:02:52.090 --> 00:02:53.490
to do this as fast as possible.

00:02:53.490 --> 00:02:55.239
I could figure out
what these numbers are,

00:02:55.239 --> 00:02:56.540
I could multiply them.

00:02:56.540 --> 00:02:59.000
But instead, let's
just take the sum.

00:02:59.000 --> 00:03:01.040
Let's just take the
sum here of this row,

00:03:01.040 --> 00:03:03.390
of this first row
that we just did.

00:03:03.390 --> 00:03:10.770
We have 1 plus 3 plus 9
plus 27, 3 plus 27 is 30,

00:03:10.770 --> 00:03:13.330
1 plus 9 is 10.

00:03:13.330 --> 00:03:16.160
So this is going to be 40.

00:03:16.160 --> 00:03:19.220
Now, whatever these numbers are,
they're all going to be 2 times

00:03:19.220 --> 00:03:19.910
these numbers.

00:03:19.910 --> 00:03:23.660
So the sum is going to be
80, and the sum over here

00:03:23.660 --> 00:03:25.390
is going to be 2 times
the previous row.

00:03:25.390 --> 00:03:28.670
Because here we multiplied by
2, here we're multiplying by 4,

00:03:28.670 --> 00:03:30.830
so it's going to be 160.

00:03:30.830 --> 00:03:33.240
And over here, we just
multiplied by 2 again,

00:03:33.240 --> 00:03:34.897
it's going to be 320.

00:03:34.897 --> 00:03:37.480
Or another way of thinking about
it, whatever the sum is here,

00:03:37.480 --> 00:03:40.450
it's going to be eight times
the sum of the first row.

00:03:40.450 --> 00:03:42.340
And I could, just so
you know what I'm doing,

00:03:42.340 --> 00:03:43.756
I could actually
put numbers here.

00:03:43.756 --> 00:03:49.410
This number would be 8, 24,
72, and whatever 8 times 27,

00:03:49.410 --> 00:03:53.730
I was at 160, 160
plus 56, so it's 216.

00:03:53.730 --> 00:03:55.120
But we don't want to do that.

00:03:55.120 --> 00:03:57.260
We just have to
think about the sums.

00:03:57.260 --> 00:04:00.710
So if you think about all of the
dividers of 27,000 that are not

00:04:00.710 --> 00:04:02.870
divisible by 5-- so
they're only divisible by 5

00:04:02.870 --> 00:04:04.910
to the zero power, I
guess you could say it.

00:04:04.910 --> 00:04:07.380
We've now figured
out their sum, it's

00:04:07.380 --> 00:04:10.990
going to be the sum
of all of these rows.

00:04:10.990 --> 00:04:13.610
So if you take 40
plus 80, you have

00:04:13.610 --> 00:04:22.430
120 plus 160 is 280
plus 320 is 600.

00:04:22.430 --> 00:04:24.290
So this is the situation.

00:04:24.290 --> 00:04:27.430
This is the sum of all of
the combinations of the 2's

00:04:27.430 --> 00:04:30.290
and the 3's that don't
have any 5's in them.

00:04:30.290 --> 00:04:31.870
Now, if you took
the same combination

00:04:31.870 --> 00:04:36.697
of 2's and 3's, so
these added up to 600,

00:04:36.697 --> 00:04:37.780
let me write it over here.

00:04:40.420 --> 00:04:45.740
So no fives Now, if you
did the same exact thing

00:04:45.740 --> 00:04:47.220
that we just did
here, but we just

00:04:47.220 --> 00:04:49.382
multiplied everything by 5.

00:04:49.382 --> 00:04:51.340
So we'd then be looking
at all the combinations

00:04:51.340 --> 00:04:54.830
that have this many twos and
this many threes, and one five,

00:04:54.830 --> 00:04:56.930
what would happen to this sum?

00:04:56.930 --> 00:04:59.730
Well, we would just
multiply it by 5.

00:04:59.730 --> 00:05:01.070
So let's multiply that by 5.

00:05:01.070 --> 00:05:06.300
So you multiply 600 by 5,
you get 30 with two zeroes,

00:05:06.300 --> 00:05:11.284
and so this is one 5 in
the prime factorization

00:05:11.284 --> 00:05:11.950
of the divisors.

00:05:15.730 --> 00:05:19.500
Now, if I wanted two 5's, I
could just multiply by 5 again.

00:05:19.500 --> 00:05:26.610
So if I multiply by 5
again, I get 15,000.

00:05:26.610 --> 00:05:29.900
This is two 5's.

00:05:29.900 --> 00:05:31.490
Another way of
thinking about this,

00:05:31.490 --> 00:05:33.730
if I just multiply
every term here

00:05:33.730 --> 00:05:37.160
by 25, which is essentially
multiplying by 5 times 5,

00:05:37.160 --> 00:05:42.120
this sum is going to be 600
times 25, which is 15,000.

00:05:42.120 --> 00:05:47.450
Now, if I have three
5's than I could just

00:05:47.450 --> 00:05:51.030
multiply this by five again.

00:05:51.030 --> 00:05:54.506
5 times 15 is 50 plus 25 is 75.

00:05:57.085 --> 00:05:59.620
So its 75,000.

00:05:59.620 --> 00:06:00.790
So now I know all the sums.

00:06:00.790 --> 00:06:02.664
If I have no 5's, the
sum of all the divisors

00:06:02.664 --> 00:06:06.096
is 600, if I have one 5,
3,000, so on and so forth.

00:06:06.096 --> 00:06:07.720
If I want the sum of
everything, I just

00:06:07.720 --> 00:06:10.320
take the sum of these numbers.

00:06:10.320 --> 00:06:12.670
Let me scroll down a little bit.

00:06:12.670 --> 00:06:16.630
So I get, well, I
have zero, zero,

00:06:16.630 --> 00:06:19.550
than in the hundreds
place, I only have a six,

00:06:19.550 --> 00:06:23.610
and then 3 plus 5 plus 5 is 13.

00:06:23.610 --> 00:06:24.350
Is that right?

00:06:24.350 --> 00:06:26.320
Yeah, that's 13.

00:06:26.320 --> 00:06:29.634
And then carry the one,
and then I have a 9.

00:06:29.634 --> 00:06:33.020
So 93,600.

00:06:33.020 --> 00:06:40.630
So the sum of all positive
divisors of 27,000, 93,600.

00:06:40.630 --> 00:06:42.800
Hope you found
that entertaining.