Taxonomy and the Tree of Life

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Kind: captions
Language: en

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This right here is a
picture of Carl Linnaeus,

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and I'm sure I'm
mispronouncing the word.

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He's a Swedish gentleman
who lived in the 1700s,

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and he's known as the
father of modern taxonomy.

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And the word
taxonomy, if you just

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split up into its
original root, it really

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is the science of really
classifying things.

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But when people talk about
taxonomy-- and in particular,

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in Carl Linnaeus'
case-- they're talking

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about the
classification of living

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things, so
classifying organisms.

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And his real innovation--
before he came about, peopled

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realize that you had
species of animals,

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that lions had certain
properties that

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made them all lions, that
they could interbreed

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and things like that, that
monkey or chimpanzees would all

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interbreed and that would
be a separate species

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and that polar bears
were separate species

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and that humans were
a separate species.

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But what he really brought
to the table was he decided,

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well, let me not just
group animals into species.

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Maybe I can group species
into other categories.

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And that's where we
get the genus from.

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You group similar
species into a genus.

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And then he went
even beyond that,

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because even the idea
of grouping things

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into a genus dated back
to the ancient Greeks.

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He said, well, why don't I
group similar genuses together

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into orders, orders together
into classes, and then classes

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together into kingdoms.

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So really what he
did is he said,

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well, maybe I can classify--
I can create a tree.

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I can create a tree of life.

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I can create a structure
so we can really

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see how far apart any
two organisms are,

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and so that's why
he's really the father

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about modern taxonomy.

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And he did not have many tools.

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All he could do was look at
his powers of observations

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and say, OK, those kind
of animals, they have fur,

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or they reproduce in this way.

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Or they lay eggs, or
they don't lay eggs.

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Or they have spinal columns, or
they don't have spinal columns.

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So that's the best that he could
do when he did his taxonomy.

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But since then,
there's obviously

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been tons of innovations
in how we perceive

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animals, or the natural world,
and our tools for studying

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them.

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So one thing that he
did not know about

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is evolution-- this
idea of common ancestry.

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And between our
understandings of evolution

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and our ability to look
back at the fossil record,

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that helps us get more
precise at figuring out

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which animals are
related to which.

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We can see, do they have a
common ancestor more recent

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or further back.

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And what even Charles
Darwin didn't have,

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which we now use as
a tool in taxonomy,

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is the genetic evidence.

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So now we don't even have to
rely on the fossil record.

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We could look at the DNA of
two species that exist today

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and see how similar is that DNA.

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And that tells us how
recently they branched apart

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if we were able to find
it in the fossil record

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or how recently in the past
did these two species become

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two different species.

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Now, with that said, I do
want to make this clear.

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And this is something
that I've always

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had a little bit-- it was fuzzy
for me the first time that I

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was exposed to this
idea of taxonomy--

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is that taxonomy is as much
an art as it's a science.

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And today, even to
this day, people

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are debating about the
best way to classify things

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and what do you
pay attention to.

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And DNA has been
the best tool so far

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in giving us a more systematic,
a more analytical way,

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of deciding how close
two animals are.

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But to a large degree, a lot
of these categories-- deciding

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where to divide along kingdom,
phylum, class, order, family,

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tribe-- these are
somewhat arbitrary.

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These are just picked based on
early taxonomists, including

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Carl Linnaeus, and
saying, well, this

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looks like a grouping
right over here.

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But they could have grouped
at a broader level or a deeper

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level.

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So these things right over
here are somewhat arbitrary.

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A more analytical way is
just to see how much DNA you

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have in common and then use that
as a measure of how far apart

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two animals are.

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Or really, I should
say, two species are,

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because this taxonomy doesn't
only apply just to animals.

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It applies to plants
and bacteria and Archaea

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and all sorts of
things, so it's actually

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a broader thing
than just animals.

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Now, with that out of
the way, what I thought

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would be fun-- just so
that we could really

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get a sense of where
modern taxonomy is, where

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the field that was essentially
fathered by Carl Linnaeus,

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where it is now, how we-- and
use that to figure out where we

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humans fit into the big picture.

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And obviously, I'm drawing
just a small fraction

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of the universe of the
organisms that we even

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know about right now.

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But at least it frames the
picture in terms of something

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we understand--
in particular, us.

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In particular, humans.

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Now, our species, we
call ourselves humans.

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But we're really Homo sapiens.

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And the sapiens is
the species part,

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and then Homo is the genus.

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And what I'm doing
right over here

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is I'm saying, well, if
Homo is the genus, what

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other species were
inside of Homo?

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And the reality is--
or at least as far

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as we know-- there are no other
living species inside of Homo.

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We probably killed them all off.

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Or maybe we interbreeded
with them somehow,

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which might have
argued that maybe they

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weren't different species.

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But more likely,
they were competing

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in the same ecosystems, and
they became endangered species

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very quickly when they
competed with our ancestors.

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But the most recent
other species

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within the genus that we know
about are the neanderthals,

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and the formal term for their
species is neanderthalensis.

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Now, if we go further
up the tree of life,

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further up the taxonomy--
and you'll sometimes

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see tribe mentioned.

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Sometimes you won't.

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And we tend to get a
little bit more granular

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the closer we get to humans.

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When we go further away
in the tree of life,

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we get a little bit
less granular sometimes.

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But that's not always
the case as well.

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You go a little bit further
up, then you get Hominini.

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And I'm sure I'm mispronouncing
some of this as well.

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But another species that's in
Homonini that is not in Homo--

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and I'm definitely not
listing all of them,

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and that's why I'm showing
all of these other branches

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over here-- is what we
call the common chimpanzee.

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And their species name
is-- their genus is Pan,

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and their species
is troglodytes.

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So you would refer to
them as Pan troglodytes.

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And that's also
another convention

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that Carl Linnaeus
came up with, is

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that you refer to a
particular species

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by its genus and
then its species.

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And you capitalize the genus,
and you lowercase a species.

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So we're Homo sapiens.

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This is Homo neanderthalensis.

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This Pan troglodytes or often
referred to as chimpanzees.

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Now, if you go up one
higher level of broadness

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on this tree of life, you
then get to the family.

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And we are in the
family Hominidae,

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and I'm sure I'm
mispronouncing it once again.

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But just to give you an
example, so everything I've

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listed so far, everything
I've talked about so far

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are within this family.

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And to show you an animal
that is not in this family,

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you just have to
look at the gorilla.

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And you could call it
the Gorillini gorilla,

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or G gorilla.

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That's its actual species name.

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And this family right over
here, sometimes the common term

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is the great apes.

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Now, you go one further
level-- and the whole reason

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why I'm doing this--
and I'm not by any means

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being exhaustive about
the other species that

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are in that family, but
that are not in our tribe.

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I'm just trying to give
you a picture of-- as we

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get further and further
out, as we get further

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out of our tribe, our
family, our order,

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we're getting to things where
the common ancestry with human

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goes further and
further back in time.

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The genetic similarities
become more and more different.

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And even just the
physical differences,

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if we look it at a very
superficial level, become

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more and more and
more different.

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So you get to even
a broader category.

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This is where you
get to the primates,

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and this is probably
something that you

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might be somewhat familiar with.

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And the term
primates is generally

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these animals that
look like they either

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live in trees or a
rain forest, or they're

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a descendant of things
that live in trees.

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So they have these things that
they can grasp things with.

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They're good at
climbing, broadly.

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Not all of them are.

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Humans are probably
the worst primates

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when it comes to climbing,
or one of the worst.

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But that's the general
classification.

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That's what we generally think
of when we think of primates.

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And if we think of a primate
that is not a great ape,

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you just have to
think of a baboon.

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So this right here is a baboon.

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It is a primate, but
it is not a great ape.

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It is probably a descendant--
some baboons actually

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don't live in trees.

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But all of them are
probably a descendant

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from things that
first lived in trees,

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and that's why their
hands and their feet

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look the way they do.

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Now you get to even a broader
level of classification.

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You get to the mammals.

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And once again,
probably something

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you're used to thinking about.

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Mammals are
air-breathing animals,

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and they tend to
have fur or hair.

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They tend to provide some
form of milk for their young.

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They have active mammary glands.

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There's other things we can
talk about, what makes a mammal.

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I'm not going to go into
the rigorous definition.

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But just to give you an
example of a mammal that is not

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a primate, I could show you
this polar bear right over here.

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This is a mammal that
is not a primate.

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And I could do other things.

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I could show you a
tiger, or I could

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show you a giraffe or a horse.

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And so by no stretch
of the imagination

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am I being comprehensive.

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But let's keep getting broader.

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Now let's go to the
class-- we're already

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at the class of Mammalia.

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Now let's go to the phylum.

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In phylum, we are--
humans and all mammals,

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we are in the phylum chordates.

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And chordates, we're
actually in the subphylum,

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which I didn't write
here, vertebrates,

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which means we have a vertebra.

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We have a spinal column
with a spinal cord in it.

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Chordates are a little
bit more general.

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Chordates is a
phylum where-- kind

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of the arrangement of
where the mouth is, where

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are the digestive organs,
where the anus is,

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where the spinal column
is, where are the brains,

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where are the eyes,
where are the mouth.

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They're kind of all
in the same place.

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And if you think
about it, everything

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I've listed here kind of has
the same general structure.

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You have a spinal column.

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You have a brain.

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You have a mouth.

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Then the mouth leads to
some type digestive column.

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And at the end of it, you
have an anus over there.

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And you have eyes in
front of the brain.

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And so this is a general way--
and I'm not being very rigorous

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here, is how you
describe a chordate.

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And to show a chordate
that is not a mammal,

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you would just have to
think of a fish or sharks.

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So this right over here
is a non-mammal chordate

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This is a great white
shark over here.

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Now, let's go even broader.

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As you'll see, now
we're going to things

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that are very, very
not human-like.

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So you go one step broader.

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Now we're in Animalia,
the kingdom of animals.

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And this is the
broadest category

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that Carl Linnaeus
thought about.

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Well actually, he did
go into trees as well.

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But when you think
of kingdom animals

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and you think of things
that aren't chordates,

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you start going into
things like insects.

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And you start going into
things like jellyfish.

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If you go even broader, now
we're talking about the domain.

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You go to Eukarya.

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So these are all
organisms that have cells.

00:10:41.790 --> 00:10:45.660
And inside those cells, they
have complex structures.

00:10:45.660 --> 00:10:48.600
So if you're a Eukarya, you have
cells with complex structures.

00:10:48.600 --> 00:10:51.470
If you're a Prokarya, you
don't have complex structures

00:10:51.470 --> 00:10:52.620
inside your cell.

00:10:52.620 --> 00:10:55.110
But other Eukarya
that are not animals

00:10:55.110 --> 00:11:00.060
include things like plants.

00:11:00.060 --> 00:11:01.610
And obviously, I'm
giving no justice

00:11:01.610 --> 00:11:03.737
to this whole branch
of the tree of life.

00:11:03.737 --> 00:11:05.820
It could be just as rich
or richer than everything

00:11:05.820 --> 00:11:06.694
I've drawn over here.

00:11:06.694 --> 00:11:10.160
This is just a small fraction
of the entire tree of life,

00:11:10.160 --> 00:11:12.240
but let's go even
broader than that.

00:11:12.240 --> 00:11:14.640
So if you go even
broader than that,

00:11:14.640 --> 00:11:17.030
you say, well, what's
a kind of life form

00:11:17.030 --> 00:11:19.300
that isn't Eukarya, that
wouldn't have these more

00:11:19.300 --> 00:11:22.760
complex cell structures, the
mitochondria in the cells,

00:11:22.760 --> 00:11:25.410
the cell nucleuses?

00:11:25.410 --> 00:11:28.602
Then you just have to think
about something like bacteria.

00:11:28.602 --> 00:11:30.060
And if you want to
go even broader,

00:11:30.060 --> 00:11:32.018
there's things like
viruses that you could even

00:11:32.018 --> 00:11:34.400
debate whether they
really even are life,

00:11:34.400 --> 00:11:36.510
because they are
dependent on other life

00:11:36.510 --> 00:11:39.060
forms for their
actual reproduction.

00:11:39.060 --> 00:11:41.860
But they do have genetic
material, like everything else.

00:11:41.860 --> 00:11:44.030
And that, to me, is kind
of a mind-blowing idea.

00:11:44.030 --> 00:11:45.580
As different as
a plant is-- look

00:11:45.580 --> 00:11:47.580
at a house plant that is
in your house right now

00:11:47.580 --> 00:11:50.410
or the tree when you walk home
or bacteria or this jellyfish.

00:11:50.410 --> 00:11:52.980
There is a commonality
in that we all have DNA.

00:11:52.980 --> 00:11:55.460
And that DNA, for the
most part, replicates

00:11:55.460 --> 00:11:57.490
in a very, very,
very, similar way.

00:11:57.490 --> 00:12:00.110
So it's actually crazy that
we actually even are related

00:12:00.110 --> 00:12:02.620
or that we even do have a
common ancestor with some

00:12:02.620 --> 00:12:03.452
of these things.

00:12:03.452 --> 00:12:05.160
And then it even begs
the question, well,

00:12:05.160 --> 00:12:06.834
what about things like viruses?

00:12:06.834 --> 00:12:08.000
Anyway, I'll leave you here.

00:12:08.000 --> 00:12:09.375
And I really just
want to let you

00:12:09.375 --> 00:12:13.310
know that-- make sure you
realize that this is a-- it's

00:12:13.310 --> 00:12:16.240
definitely worth studying,
because we understand where we

00:12:16.240 --> 00:12:19.735
fit in in the universe
of living things.

00:12:19.735 --> 00:12:21.360
But I also want to
let you know that it

00:12:21.360 --> 00:12:23.510
is a little bit of an
art on where you decide

00:12:23.510 --> 00:12:25.930
where to make these
classifications

00:12:25.930 --> 00:12:29.000
or where you decide to
focus on, whether you

00:12:29.000 --> 00:12:31.050
want to focus on
what properties,

00:12:31.050 --> 00:12:33.850
whether it's how they reproduce
or how they feed their young

00:12:33.850 --> 00:12:37.780
or can they move around
or what they breathe

00:12:37.780 --> 00:12:39.200
or whatever, things like that.

00:12:39.200 --> 00:12:41.500
Anyway, I'll let you go there.


Species

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.985
We humans like to get
our heads around all

00:00:02.985 --> 00:00:06.400
of the complexity around
us by classifying things.

00:00:06.400 --> 00:00:09.520
And you could imagine
there's no more obvious thing

00:00:09.520 --> 00:00:12.510
to classify than all of the
living things around us,

00:00:12.510 --> 00:00:14.441
than all of the life
that surrounds us.

00:00:14.441 --> 00:00:16.149
So what I want to
start talking about is,

00:00:16.149 --> 00:00:19.060
how do we classify all
of the life around us?

00:00:19.060 --> 00:00:22.620
And this is more often
generally referred to taxonomy.

00:00:22.620 --> 00:00:24.149
But the most basic
question you have

00:00:24.149 --> 00:00:25.690
when you look at
all the life around,

00:00:25.690 --> 00:00:28.930
you start to see similarities
between some of these living

00:00:28.930 --> 00:00:29.710
things.

00:00:29.710 --> 00:00:33.560
You see, obviously this
thing right over here

00:00:33.560 --> 00:00:36.670
is more similar to the
things that look like it,

00:00:36.670 --> 00:00:40.170
than it does to the grass
behind it, or to that tree.

00:00:40.170 --> 00:00:43.190
And so we start saying, well
maybe I should group this thing

00:00:43.190 --> 00:00:46.580
right over here into a group
with other things like it.

00:00:46.580 --> 00:00:49.740
And that very most
building block

00:00:49.740 --> 00:00:53.240
of how we classify all of
the living things around us,

00:00:53.240 --> 00:00:56.320
is putting them into
buckets called species.

00:00:59.190 --> 00:01:02.110
So for example, this is
one particular animal,

00:01:02.110 --> 00:01:04.840
but we see other animals
that seem to look like it,

00:01:04.840 --> 00:01:08.910
and so we say they're all
part of the species of lions.

00:01:08.910 --> 00:01:11.347
And this animal,
it's one animal,

00:01:11.347 --> 00:01:13.180
and there's other animals
that have stripes,

00:01:13.180 --> 00:01:15.470
but some might be fatter,
or taller, or skinnier,

00:01:15.470 --> 00:01:17.324
or whatever else,
darker, or lighter,

00:01:17.324 --> 00:01:18.740
but we say they're
similar enough,

00:01:18.740 --> 00:01:22.352
that we call them all tigers.

00:01:22.352 --> 00:01:24.060
We call all the animals,
even though they

00:01:24.060 --> 00:01:26.665
might be a little bit bigger,
or skinnier, or fatter,

00:01:26.665 --> 00:01:30.200
or lighter, or darker, we
call all of them-- they look

00:01:30.200 --> 00:01:31.860
similar to this thing
right over here--

00:01:31.860 --> 00:01:36.727
we would call this a donkey.

00:01:36.727 --> 00:01:38.310
We would call the
things that we think

00:01:38.310 --> 00:01:41.920
are like this animal
right here, a horse.

00:01:41.920 --> 00:01:45.340
Now, that might seem like a
pretty straightforward way

00:01:45.340 --> 00:01:46.090
to think about it.

00:01:46.090 --> 00:01:48.340
Oh, everything that looks kind
of like this character right

00:01:48.340 --> 00:01:49.798
here is a lion,
anything that looks

00:01:49.798 --> 00:01:52.100
kind of like this character
right here is a tiger.

00:01:52.100 --> 00:01:55.810
But that, by itself, is not
a good enough definition

00:01:55.810 --> 00:01:56.560
for a species.

00:01:56.560 --> 00:01:58.579
Things that look like
each other or things that

00:01:58.579 --> 00:02:00.370
act like each other,
because what we'll see

00:02:00.370 --> 00:02:02.330
is that there's some
things that could

00:02:02.330 --> 00:02:05.330
be very different, at least
in and how they look or act,

00:02:05.330 --> 00:02:06.940
but are actually
closely related.

00:02:06.940 --> 00:02:09.639
And we'll talk about what it
means to be closely related.

00:02:09.639 --> 00:02:12.180
And then we can see things that
look very similar, that they

00:02:12.180 --> 00:02:15.010
have similar structures or
they have similar behavior,

00:02:15.010 --> 00:02:18.130
like for example bats and
birds, but they are actually

00:02:18.130 --> 00:02:20.660
all very, very
distantly related.

00:02:20.660 --> 00:02:23.220
So we need a more exact
definition for species

00:02:23.220 --> 00:02:25.410
than just things that
look like each other,

00:02:25.410 --> 00:02:27.500
or just things that
act like each other.

00:02:27.500 --> 00:02:30.280
And so the most typical
definition for species

00:02:30.280 --> 00:02:34.300
are animals that can interbreed.

00:02:39.030 --> 00:02:41.110
And when we say
interbreed, literally they

00:02:41.110 --> 00:02:43.030
can produce offspring
with each other,

00:02:43.030 --> 00:02:45.960
and the offspring are fertile.

00:02:50.060 --> 00:02:54.400
Which means that the offspring
can then further have babies,

00:02:54.400 --> 00:02:56.550
that they're not
sterile, that they're

00:02:56.550 --> 00:02:58.820
capable of breeding
with other animals

00:02:58.820 --> 00:03:01.570
and producing more offspring.

00:03:01.570 --> 00:03:04.690
And to show an example of this,
this right here is a male lion.

00:03:04.690 --> 00:03:08.190
You find a male lion
and a female lioness

00:03:08.190 --> 00:03:11.720
and most of the time they will
be able to have offspring,

00:03:11.720 --> 00:03:14.180
and those offspring
can go and mate

00:03:14.180 --> 00:03:17.770
with other lions or lionesses,
depending on their sex,

00:03:17.770 --> 00:03:19.620
and then they can
have viable offspring.

00:03:19.620 --> 00:03:22.080
So it seems to work out
pretty well for lions.

00:03:22.080 --> 00:03:24.350
Same thing is true of tigers.

00:03:24.350 --> 00:03:26.590
Now, it does turn
out that if you

00:03:26.590 --> 00:03:35.720
get a male lion and a female
tigress they can breed.

00:03:35.720 --> 00:03:38.719
They can breed and they
can produce offspring.

00:03:38.719 --> 00:03:40.510
And their offspring--
which was made famous

00:03:40.510 --> 00:03:44.010
by Napoleon Dynamite, he
was kind of fascinated by,

00:03:44.010 --> 00:03:45.800
these are kind of
fascinating animals.

00:03:45.800 --> 00:03:48.210
Their offspring
is called a liger.

00:03:48.210 --> 00:03:51.140
You get a male lion
breeding with a female tiger

00:03:51.140 --> 00:03:53.510
you produce a liger,
which is a hybrid,

00:03:53.510 --> 00:03:55.847
it's a cross between
a lion and a tiger.

00:03:55.847 --> 00:03:57.180
And they're fascinating animals.

00:03:57.180 --> 00:03:59.810
They're actually larger
than either lions or tigers.

00:03:59.810 --> 00:04:02.530
They are the largest
cats that we know of.

00:04:02.530 --> 00:04:06.830
But these ligers cannot be
referred as a separate species.

00:04:06.830 --> 00:04:09.140
Or you can't say
that lions and tigers

00:04:09.140 --> 00:04:12.020
are the same species,
because even though they

00:04:12.020 --> 00:04:16.550
are able to interbreed, their
offspring, for the most part,

00:04:16.550 --> 00:04:20.410
is not fertile, is not
able to produce offspring.

00:04:20.410 --> 00:04:22.360
There have been one off
stories about ligers

00:04:22.360 --> 00:04:25.451
being mated with either
a lion or a tiger,

00:04:25.451 --> 00:04:26.700
but those are one off stories.

00:04:26.700 --> 00:04:29.280
In general, ligers
can't interbreed.

00:04:29.280 --> 00:04:32.400
And in general, this
combination isn't

00:04:32.400 --> 00:04:35.140
going to produce offspring
that can keep interbreeding

00:04:35.140 --> 00:04:36.210
or that are fertile.

00:04:36.210 --> 00:04:38.770
So that's why we say
that lions and tigers are

00:04:38.770 --> 00:04:40.309
different species.

00:04:40.309 --> 00:04:42.850
And that liger, we wouldn't even
call it as a species at all.

00:04:42.850 --> 00:04:46.130
We would actually call it a
hybrid between two species.

00:04:46.130 --> 00:04:48.005
Now the same thing is
true-- and actually you

00:04:48.005 --> 00:04:49.505
might be asking
yourself, well, this

00:04:49.505 --> 00:04:51.729
was a male lion and
a female tigress,

00:04:51.729 --> 00:04:53.270
what if we went the
other way around?

00:04:53.270 --> 00:04:57.332
What if we had a female
lioness and a male tiger?

00:04:57.332 --> 00:04:59.540
In that case, you would
produce something else called

00:04:59.540 --> 00:05:01.500
a tiglon or a tiglon,
I actually don't

00:05:01.500 --> 00:05:02.880
know how to pronounce that.

00:05:02.880 --> 00:05:04.400
And that is a
different hybrid that

00:05:04.400 --> 00:05:08.370
has slightly different
properties than a liger.

00:05:08.370 --> 00:05:11.930
I encourage you to look
up what a tiglon is.

00:05:11.930 --> 00:05:16.990
Similarly, you give
me a male donkey.

00:05:16.990 --> 00:05:19.760
And donkeys are clearly
a species by themselves,

00:05:19.760 --> 00:05:22.680
because if you give me a male
donkey and a female donkey they

00:05:22.680 --> 00:05:24.630
can reproduce, produce
another donkey,

00:05:24.630 --> 00:05:27.140
and then that donkey can
mate with other donkeys

00:05:27.140 --> 00:05:28.550
to produce more
and more donkeys.

00:05:28.550 --> 00:05:31.160
So not only can a donkey
interbreed with another donkey,

00:05:31.160 --> 00:05:33.750
but that product,
that child donkey,

00:05:33.750 --> 00:05:37.300
can keep interbreeding
with other donkeys.

00:05:37.300 --> 00:05:39.910
Similarly, horses, they
can interbreed and produce

00:05:39.910 --> 00:05:41.090
fertile offspring.

00:05:41.090 --> 00:05:46.890
But if you give me a male
donkey and a female horse

00:05:46.890 --> 00:05:49.940
they can mate and they
can produce a mule.

00:05:53.050 --> 00:05:56.350
But once again, like
the ligers, mules

00:05:56.350 --> 00:05:58.760
are not, at least
as far as I know,

00:05:58.760 --> 00:06:01.020
mules they're not fertile.

00:06:01.020 --> 00:06:02.860
Mules cannot produce
further offspring.

00:06:02.860 --> 00:06:05.000
They cannot interbreed
with each other.

00:06:05.000 --> 00:06:08.440
And because, even though
donkeys and horses can

00:06:08.440 --> 00:06:11.900
breed and produce mules, their
offspring aren't fertile.

00:06:11.900 --> 00:06:14.110
We don't consider
donkeys and horses

00:06:14.110 --> 00:06:15.930
part of the same species.

00:06:15.930 --> 00:06:19.310
And we would consider mules,
like a liger or a tiglon,

00:06:19.310 --> 00:06:21.910
we would consider them a hybrid.

00:06:21.910 --> 00:06:25.402
So these are all hybrids,
or we would call cross.

00:06:25.402 --> 00:06:26.860
In general, the
word hybrid is used

00:06:26.860 --> 00:06:29.160
when you have two things,
two different types that

00:06:29.160 --> 00:06:32.990
are somehow coming together,
somehow having a combination.

00:06:32.990 --> 00:06:34.920
And once again, like the
case with the tiglon,

00:06:34.920 --> 00:06:36.336
you might say,
well, what if I had

00:06:36.336 --> 00:06:40.415
a female donkey
and a male horse?

00:06:40.415 --> 00:06:42.290
And then you would
actually produce something

00:06:42.290 --> 00:06:48.539
called a hinny, which
isn't as common as a mule.

00:06:48.539 --> 00:06:50.080
And people like to
use mules, they're

00:06:50.080 --> 00:06:52.520
actually very good work
animals because they

00:06:52.520 --> 00:06:55.080
have some of the good properties
of both donkeys and horses.

00:06:55.080 --> 00:06:57.790
Hinnies are less common, but
once again, it is possible.

00:06:57.790 --> 00:07:00.840
And they have different
properties than mules.

00:07:00.840 --> 00:07:03.122
And I do want to
emphasize this idea.

00:07:03.122 --> 00:07:04.830
Because when we started
off we just tried

00:07:04.830 --> 00:07:06.871
to think about, well, how
can we classify things?

00:07:06.871 --> 00:07:09.420
And we said, hey, maybe things
that look and act similar,

00:07:09.420 --> 00:07:12.056
we can call a species.

00:07:12.056 --> 00:07:13.930
And maybe things that
look and act different,

00:07:13.930 --> 00:07:15.340
we shouldn't call them species.

00:07:15.340 --> 00:07:18.070
But I want to show you a
very typical case, one that's

00:07:18.070 --> 00:07:21.760
really all around us all the
time, where this definition--

00:07:21.760 --> 00:07:25.190
animals that can interbreed
and the offspring are fertile--

00:07:25.190 --> 00:07:28.410
really does seem to become much,
much more important than just

00:07:28.410 --> 00:07:30.840
some notion of animals
that look alike

00:07:30.840 --> 00:07:33.120
or animals that act the same.

00:07:33.120 --> 00:07:37.220
And the best example
of that is with dogs.

00:07:37.220 --> 00:07:39.820
As I said, this is a very
typical species here,

00:07:39.820 --> 00:07:42.800
because dogs-- and I just
took a sample of some

00:07:42.800 --> 00:07:45.710
of the different types
of breeds of dogs--

00:07:45.710 --> 00:07:48.080
they can look very,
very different.

00:07:48.080 --> 00:07:50.760
It's obvious, look at the
difference between these dogs.

00:07:50.760 --> 00:07:52.220
For example, this
little chihuahua

00:07:52.220 --> 00:07:53.770
here and this dog
right over here.

00:07:53.770 --> 00:07:56.040
Obviously, they're
size-wise, their look,

00:07:56.040 --> 00:07:59.750
and even how they act are much,
much more different than maybe

00:07:59.750 --> 00:08:02.040
how this donkey would act
relative to this horse,

00:08:02.040 --> 00:08:04.820
or how this lion would act
relative to this tigress.

00:08:04.820 --> 00:08:07.739
And they obviously
look very different,

00:08:07.739 --> 00:08:09.280
they have completely
different sizes,

00:08:09.280 --> 00:08:12.899
but these two things
actually can interbreed--

00:08:12.899 --> 00:08:14.440
although for these
two in particular,

00:08:14.440 --> 00:08:16.780
it seems like the mechanics
would get kind of difficult--

00:08:16.780 --> 00:08:18.850
but assuming they get over
the mechanical hurdles

00:08:18.850 --> 00:08:21.510
they could interbreed and
produce fertile offspring.

00:08:21.510 --> 00:08:25.950
Same for these two characters,
same for these two characters

00:08:25.950 --> 00:08:26.735
over here.

00:08:26.735 --> 00:08:29.110
And because of that, even
though all the different breeds

00:08:29.110 --> 00:08:30.790
of dogs-- and most
of this is really

00:08:30.790 --> 00:08:33.400
due to humans' doings
of trying to breed

00:08:33.400 --> 00:08:36.950
for a specific traits-- even
though they look so different,

00:08:36.950 --> 00:08:39.440
and even though they act so
different, because they can

00:08:39.440 --> 00:08:42.190
interbreed, and they could
produce fertile offspring,

00:08:42.190 --> 00:08:44.740
we consider all of
these things to be

00:08:44.740 --> 00:08:47.730
members of the same species.


Converting repeating decimals to fractions 2

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:02.050
In the last video,
we did some examples

00:00:02.050 --> 00:00:04.670
where we had one digit
repeating on and on forever,

00:00:04.670 --> 00:00:06.750
and we were able to convert
those into fractions.

00:00:06.750 --> 00:00:09.540
In this video, we want to tackle
something a little bit more

00:00:09.540 --> 00:00:11.670
interesting, which is
multiple digits repeating

00:00:11.670 --> 00:00:13.150
on and on forever.

00:00:13.150 --> 00:00:19.000
So let's say I had
0.36 repeating, which

00:00:19.000 --> 00:00:22.130
is the same thing as 0 point--
since the bar's over the 3

00:00:22.130 --> 00:00:25.920
and the 6, both of
those repeat-- 363636.

00:00:25.920 --> 00:00:29.720
And it just keeps going on and
on and on like that forever.

00:00:29.720 --> 00:00:31.560
Now the key to doing
this type of problem

00:00:31.560 --> 00:00:34.360
is, so like we did
in the last video,

00:00:34.360 --> 00:00:35.800
we set this as equal to x.

00:00:35.800 --> 00:00:37.990
And instead of just
multiplying it by 10-- 10

00:00:37.990 --> 00:00:40.070
would only shift
it one over-- we

00:00:40.070 --> 00:00:45.370
want to shift it over enough
so that when we line them up,

00:00:45.370 --> 00:00:48.440
the decimal parts will still
line up with each other.

00:00:48.440 --> 00:00:50.430
And to do that we,
want to actually shift

00:00:50.430 --> 00:00:52.720
the decimal space
two to the right.

00:00:52.720 --> 00:00:55.070
And to shift it
two to the right,

00:00:55.070 --> 00:00:59.030
we have to have multiplied by
100 or 10 to the second power.

00:00:59.030 --> 00:01:02.460
So 100x is going to
be equal to what?

00:01:02.460 --> 00:01:05.660
We're shifting this two
to the right-- one, two.

00:01:05.660 --> 00:01:11.100
So 100x is going to be equal
to-- the decimal is going

00:01:11.100 --> 00:01:17.230
to be there now, so it's
going to be 36.363636

00:01:17.230 --> 00:01:18.850
on and on and on forever.

00:01:18.850 --> 00:01:20.350
And then let me
rewrite x over here.

00:01:20.350 --> 00:01:22.550
We're going to subtract
that from the 100x.

00:01:22.550 --> 00:01:29.690
x is equal to 0.363636
repeating on and on forever.

00:01:29.690 --> 00:01:33.315
And notice when we multiplied by
100x, the 3's and the 6's still

00:01:33.315 --> 00:01:35.440
line up with each other
when we align the decimals.

00:01:35.440 --> 00:01:36.815
And you want to
make sure you get

00:01:36.815 --> 00:01:39.402
the decimals lined
up appropriately.

00:01:39.402 --> 00:01:40.860
And the reason why
this is valuable

00:01:40.860 --> 00:01:43.350
is now that when we
subtract x from 100x,

00:01:43.350 --> 00:01:46.000
the repeating parts
will cancel out.

00:01:46.000 --> 00:01:47.840
So let's subtract.

00:01:47.840 --> 00:01:50.170
Let us subtract
these two things.

00:01:50.170 --> 00:01:53.170
So on the left-hand side,
we have 100x minus x.

00:01:53.170 --> 00:01:56.100
So that gives us 99x.

00:01:56.100 --> 00:01:58.650
And then we get, on
the right-hand side,

00:01:58.650 --> 00:02:00.300
this part cancels
out with that part.

00:02:00.300 --> 00:02:03.930
And we're just left with 36.

00:02:03.930 --> 00:02:07.900
We can divide both
sides by 99, and we

00:02:07.900 --> 00:02:14.240
are left with x is
equal to 36 over 99.

00:02:14.240 --> 00:02:19.110
And both the numerator and the
denominator is divisible by 9,

00:02:19.110 --> 00:02:20.390
so we can reduce this.

00:02:20.390 --> 00:02:22.930
If we divide the
numerator by 9, we get 4.

00:02:22.930 --> 00:02:25.280
The denominator
by 9-- we get 11.

00:02:25.280 --> 00:02:31.760
So 0.363636 forever and
forever repeating is 4/11.

00:02:31.760 --> 00:02:33.960
Now let's do another
interesting one.

00:02:33.960 --> 00:02:48.340
Let's say we have the number
0.714, and the 14 is repeating.

00:02:48.340 --> 00:02:50.110
And so this is the same thing.

00:02:50.110 --> 00:02:52.160
So notice, the 714
isn't going to repeat.

00:02:52.160 --> 00:02:54.250
Just the 14 is going to repeat.

00:02:54.250 --> 00:03:00.690
So this is 0.7141414,
on and on and on and on.

00:03:00.690 --> 00:03:02.910
So let's set this equal to x.

00:03:02.910 --> 00:03:06.330
Now you might be tempted to
multiply this by 1,000x to get

00:03:06.330 --> 00:03:12.556
the decimal all the
way clear of 714.

00:03:12.556 --> 00:03:14.180
But you actually
don't want to do that.

00:03:14.180 --> 00:03:17.670
You want to shift it just enough
so that the repeating pattern

00:03:17.670 --> 00:03:21.751
can be right under itself
when you do the subtraction.

00:03:21.751 --> 00:03:23.500
So again in this
situation, even though we

00:03:23.500 --> 00:03:25.640
have three numbers
behind the decimal point,

00:03:25.640 --> 00:03:27.390
because only two of
them are repeating,

00:03:27.390 --> 00:03:30.350
we only want to multiply
by 10 to the second power.

00:03:30.350 --> 00:03:33.010
So once again, you want
to multiply by 100.

00:03:33.010 --> 00:03:37.250
So you get 100x is equal to--
we're moving the decimal two

00:03:37.250 --> 00:03:39.680
to the right, one,
two-- so it's going

00:03:39.680 --> 00:03:44.360
to be 71.4141, on
and on and on and on.

00:03:44.360 --> 00:03:52.810
So it's going to be 71.4141414
and on and on and on.

00:03:52.810 --> 00:03:54.770
And then let me rewrite
x right below this.

00:03:54.770 --> 00:04:00.380
We have x is equal to 0.7141414.

00:04:00.380 --> 00:04:03.340
And notice, now the
141414's, they're

00:04:03.340 --> 00:04:05.040
lined up right below each other.

00:04:05.040 --> 00:04:08.190
So it's going to work
out when we subtract.

00:04:08.190 --> 00:04:10.170
So let's subtract these things.

00:04:10.170 --> 00:04:14.230
100x minus x is 99x.

00:04:14.230 --> 00:04:17.750
And this is going to be equal
to-- these 1414's are going

00:04:17.750 --> 00:04:20.600
to cancel with those 1414's.

00:04:20.600 --> 00:04:22.800
And we have 71.4 minus 0.7.

00:04:22.800 --> 00:04:25.860
And we can do this in our head,
or we can borrow if you like.

00:04:25.860 --> 00:04:27.330
This could be a 14.

00:04:27.330 --> 00:04:28.620
This is a 0.

00:04:28.620 --> 00:04:34.310
So you have 14 minus 7
is 7 and then 70 minus 0.

00:04:34.310 --> 00:04:39.260
So you have 99x
is equal to 70.7.

00:04:39.260 --> 00:04:41.300
And then we can divide
both sides by 99.

00:04:41.300 --> 00:04:43.110
And you could see all of
the sudden something strange

00:04:43.110 --> 00:04:44.985
is happening because we
still have a decimal.

00:04:44.985 --> 00:04:46.470
But we can fix
that up at the end.

00:04:46.470 --> 00:04:48.145
So let's divide
both sides by 99.

00:04:51.380 --> 00:04:55.534
You get x is equal
to 70.7 over 99.

00:04:55.534 --> 00:04:57.200
Now obviously, we
haven't converted this

00:04:57.200 --> 00:04:58.300
into a pure fraction yet.

00:04:58.300 --> 00:05:00.190
We still have a decimal
in the numerator.

00:05:00.190 --> 00:05:01.840
But that's pretty easy to fix.

00:05:01.840 --> 00:05:05.170
You just have to multiply the
numerator and denominator by 10

00:05:05.170 --> 00:05:06.320
to get rid of this decimal.

00:05:06.320 --> 00:05:09.270
So let's multiply the numerator
by 10 and the denominator

00:05:09.270 --> 00:05:10.280
by 10.

00:05:10.280 --> 00:05:19.780
And so we get 707/990.

00:05:19.780 --> 00:05:21.640
Let's do one more
example over here.

00:05:21.640 --> 00:05:26.810
So let's say we had something
like-- let me write this way--

00:05:26.810 --> 00:05:35.710
3.257 repeating, and we want to
convert this into a fraction.

00:05:35.710 --> 00:05:39.080
So once again, we
set this equal to x.

00:05:39.080 --> 00:05:42.400
And notice, this is
going to be 3.257257257.

00:05:42.400 --> 00:05:44.730
The 257 is going to
repeat on and on and on.

00:05:44.730 --> 00:05:46.690
Since we have three
digits that are repeating,

00:05:46.690 --> 00:05:52.660
we want to think about 1,000x,
10 to the third power times x.

00:05:52.660 --> 00:05:54.560
And that'll let us
shift it just right

00:05:54.560 --> 00:05:57.080
so that the repeating
parts can cancel out.

00:05:57.080 --> 00:06:02.157
So 1,000x is going
to be equal to what?

00:06:02.157 --> 00:06:03.740
We're going to shift
the decimal three

00:06:03.740 --> 00:06:06.500
to the right-- one, two, three.

00:06:06.500 --> 00:06:10.490
So it's going to
be 3,257 point--

00:06:10.490 --> 00:06:12.180
and then the 257
keeps repeating.

00:06:12.180 --> 00:06:18.390
257257257 keeps going on
and on and on forever.

00:06:18.390 --> 00:06:20.850
And then we're going to
subtract x from that.

00:06:20.850 --> 00:06:23.231
So here is x. x is equal to 3.

00:06:23.231 --> 00:06:25.480
You want to make sure you
have your decimals lined up.

00:06:25.480 --> 00:06:33.060
It's 3.257257257 dot, dot,
dot-- keeps going on forever.

00:06:33.060 --> 00:06:35.100
And notice, when we
multiplied it by 1,000,

00:06:35.100 --> 00:06:38.690
it allowed us to line up the
257's so that when we subtract,

00:06:38.690 --> 00:06:40.790
the repeating part cancels out.

00:06:40.790 --> 00:06:43.050
So let's do that subtraction.

00:06:43.050 --> 00:06:45.004
On the left-hand side,
1,000 of something

00:06:45.004 --> 00:06:46.420
minus 1 of that
something-- you're

00:06:46.420 --> 00:06:50.590
left with 999 of that something.

00:06:50.590 --> 00:06:53.110
This part is going to
cancel out with that part.

00:06:53.110 --> 00:06:57.650
It's going to be equal to--
let's see, 7 minus 3 is 4.

00:06:57.650 --> 00:07:01.810
And then you have the
5, the 2, and the 3.

00:07:01.810 --> 00:07:07.240
So you get 999x
is equal to 3,254.

00:07:07.240 --> 00:07:15.830
And then you can divide
both sides of this by 999.

00:07:15.830 --> 00:07:23.590
And you are left with x
is equal to 3,254/999.

00:07:23.590 --> 00:07:25.590
And so obviously, this
is an improper fraction.

00:07:25.590 --> 00:07:27.690
The numerator is larger
than the denominator.

00:07:27.690 --> 00:07:30.414
You could convert this to a
proper fraction if you like.

00:07:30.414 --> 00:07:31.830
One way, you could
have just tried

00:07:31.830 --> 00:07:36.820
to figure out what to the 0.257
repeating forever is equal to

00:07:36.820 --> 00:07:38.540
and just had the 3
being the whole number

00:07:38.540 --> 00:07:39.600
part of a mixed fraction.

00:07:39.600 --> 00:07:43.460
Or you could just
divide 999 into 3,254.

00:07:43.460 --> 00:07:45.690
Actually, we could do that
pretty straightforwardly.

00:07:45.690 --> 00:07:48.290
It goes into it three times,
and the remainder-- well,

00:07:48.290 --> 00:07:53.010
let me just do it, just
to go through the motions.

00:07:53.010 --> 00:07:56.660
So 999 goes into 3,254.

00:07:56.660 --> 00:07:58.400
It'll go into it three times.

00:07:58.400 --> 00:08:01.160
And we know that because
this is originally 3.257,

00:08:01.160 --> 00:08:02.910
so we're just going
to find the remainder.

00:08:02.910 --> 00:08:04.645
So 3 times 9 is 27.

00:08:08.020 --> 00:08:10.450
But we have to add
the 2, so it's 29.

00:08:10.450 --> 00:08:12.650
3 times 9 is 27.

00:08:12.650 --> 00:08:15.040
We have a 2, so it's 29.

00:08:15.040 --> 00:08:17.310
And so we are left
with, if we subtract,

00:08:17.310 --> 00:08:21.220
if we regroup or borrow or
however we want to call it,

00:08:21.220 --> 00:08:22.860
this could be a 14.

00:08:22.860 --> 00:08:24.201
And then this could be a 4.

00:08:24.201 --> 00:08:25.450
Let me do this in a new color.

00:08:29.000 --> 00:08:31.490
And then the 4 is still
smaller than this 9,

00:08:31.490 --> 00:08:32.770
so we need to regroup again.

00:08:32.770 --> 00:08:36.240
So then this could be a 14,
and then this could be a 1.

00:08:36.240 --> 00:08:38.020
But this is smaller
than this 9 right

00:08:38.020 --> 00:08:40.000
over here, so we regroup again.

00:08:40.000 --> 00:08:42.780
This would be an 11,
and then this is a 2.

00:08:42.780 --> 00:08:45.580
14 minus 7 is 7.

00:08:45.580 --> 00:08:48.782
14 minus 9 is 5.

00:08:48.782 --> 00:08:51.740
11 minus 9 is 2.

00:08:51.740 --> 00:08:55.520
So we are left with--
did I do that right?

00:08:55.520 --> 00:09:03.580
Yep-- so this is going to
be equal to 3 and 257/999.

00:09:03.580 --> 00:09:05.520
And we're done.


Area of a parallelogram

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:05.380
We know that quadrilateral ABCD
over here is a parallelogram.

00:00:05.380 --> 00:00:07.540
And what I want to
discuss in this video

00:00:07.540 --> 00:00:10.980
is a general way of finding
the area of a parallelogram.

00:00:10.980 --> 00:00:13.460
In the last video, we talked
about a particular way

00:00:13.460 --> 00:00:15.569
of finding the
area of a rhombus.

00:00:15.569 --> 00:00:17.610
You could take half the
product of its diagonals.

00:00:17.610 --> 00:00:19.610
And a rhombus is
a parallelogram.

00:00:19.610 --> 00:00:22.389
But you can't just
generally take

00:00:22.389 --> 00:00:24.680
half the product of the
diagonals of any parallelogram.

00:00:24.680 --> 00:00:26.070
It has to be a rhombus.

00:00:26.070 --> 00:00:28.520
And now we're just going to
talk about parallelograms.

00:00:28.520 --> 00:00:30.690
So what do we know
about parallelograms?

00:00:30.690 --> 00:00:33.150
Well, we know the opposite
sides are parallel.

00:00:33.150 --> 00:00:35.350
So that side is
parallel to that side,

00:00:35.350 --> 00:00:37.630
and this side is
parallel to this side.

00:00:37.630 --> 00:00:40.200
And we also know that
opposite sides are congruent.

00:00:40.200 --> 00:00:42.490
So this length is
equal to this length,

00:00:42.490 --> 00:00:47.050
and this length is equal
to this length over here.

00:00:47.050 --> 00:00:52.190
Now, if we draw a diagonal--
I'll draw a diagonal AC--

00:00:52.190 --> 00:00:55.940
we can split our parallelogram
into two triangles.

00:00:55.940 --> 00:00:57.610
And we've proven
this multiple times.

00:00:57.610 --> 00:00:59.230
These two triangles
are congruent.

00:00:59.230 --> 00:01:02.100
But we can do it in a
pretty straightforward way.

00:01:02.100 --> 00:01:03.240
We can look.

00:01:03.240 --> 00:01:06.730
Obviously, AD is equal to BC.

00:01:06.730 --> 00:01:09.510
We have DC is equal to AB.

00:01:09.510 --> 00:01:12.150
And then both of these
triangles share this third side

00:01:12.150 --> 00:01:13.220
right over here.

00:01:13.220 --> 00:01:16.450
They both share AC.

00:01:16.450 --> 00:01:20.320
So we can say triangle-- let me
write this in yellow-- we could

00:01:20.320 --> 00:01:25.830
say triangle ADC is
congruent to triangle-- let

00:01:25.830 --> 00:01:26.580
me get this right.

00:01:26.580 --> 00:01:29.590
So it's going to be
congruent to triangle--

00:01:29.590 --> 00:01:34.040
so I said A, D, C. So I went
along this double magenta

00:01:34.040 --> 00:01:39.040
slash first, then the
pink, and then I went D,

00:01:39.040 --> 00:01:41.310
and then I went the last one.

00:01:41.310 --> 00:01:44.075
So I'm going to say CBA.

00:01:44.075 --> 00:01:45.950
Because I went the double
magenta, then pink,

00:01:45.950 --> 00:01:46.840
then the last one.

00:01:46.840 --> 00:01:50.500
So CBA, triangle CBA.

00:01:50.500 --> 00:01:55.220
And this is by side,
side, side congruency.

00:01:55.220 --> 00:01:57.400
All three sides, they have
three corresponding sides

00:01:57.400 --> 00:01:58.340
that are congruent
to each other.

00:01:58.340 --> 00:02:00.620
So the triangles are
congruent to each other.

00:02:00.620 --> 00:02:03.270
And what that tells us is
that the areas of these two

00:02:03.270 --> 00:02:05.300
triangles are going
to be the same.

00:02:05.300 --> 00:02:10.960
So if I want to find the area of
ABCD, the whole parallelogram,

00:02:10.960 --> 00:02:13.920
it's going to be equal
to the area of triangle--

00:02:13.920 --> 00:02:18.530
let me just write it here--
it's equal to the area of ADC

00:02:18.530 --> 00:02:21.790
plus the area of CBA.

00:02:21.790 --> 00:02:27.140
But the area of CBA is just the
same thing as the area of ADC

00:02:27.140 --> 00:02:30.140
because they are congruent,
by side, side, side.

00:02:30.140 --> 00:02:34.980
So this is just going to be two
times the area of triangle ADC.

00:02:34.980 --> 00:02:36.760
Which is convenient
for us, because we

00:02:36.760 --> 00:02:40.110
know how to find the
areas of triangles.

00:02:40.110 --> 00:02:43.430
Area of triangles is literally
just 1/2 times base times

00:02:43.430 --> 00:02:44.280
height.

00:02:44.280 --> 00:02:49.440
So it's 1/2 times base times
height of this triangle.

00:02:49.440 --> 00:02:52.170
And we are given
the base of ADC.

00:02:52.170 --> 00:02:54.000
It is this length
right over here.

00:02:54.000 --> 00:02:55.052
It is DC.

00:02:55.052 --> 00:02:57.831
You could view it as the base
of the entire parallelogram.

00:02:57.831 --> 00:02:59.580
And if we wanted to
figure out the height,

00:02:59.580 --> 00:03:02.860
we could draw an
altitude down like this.

00:03:02.860 --> 00:03:05.320
So this is perpendicular.

00:03:05.320 --> 00:03:08.340
We could call that the
height right over there.

00:03:08.340 --> 00:03:14.620
So if you want the total
area of parallelogram ABCD,

00:03:14.620 --> 00:03:18.330
it is equal to two times one
half times base times height.

00:03:18.330 --> 00:03:20.285
Well, two times one
half is just one.

00:03:20.285 --> 00:03:23.290
And so you're just left
with base times height.

00:03:23.290 --> 00:03:24.610
So we can call this b.

00:03:24.610 --> 00:03:30.880
So it's just b times this height
over here, base times height.

00:03:30.880 --> 00:03:31.900
So that's a neat result.

00:03:31.900 --> 00:03:34.340
And you might have guessed
that this would be the case.

00:03:34.340 --> 00:03:36.590
But if you want to find the
area of any parallelogram,

00:03:36.590 --> 00:03:38.460
and if you can figure
out the height,

00:03:38.460 --> 00:03:40.550
it is literally you just
take one of the bases,

00:03:40.550 --> 00:03:43.862
because both bases are going
to be the-- opposite sides are

00:03:43.862 --> 00:03:46.320
equal, so it could have been
either that side or that side,

00:03:46.320 --> 00:03:47.370
times the height.

00:03:47.370 --> 00:03:49.369
So that's one way you
could have found the area.

00:03:49.369 --> 00:03:50.589
Or you could have multiplied.

00:03:50.589 --> 00:03:53.130
The other way to think about it
is you could have multiplied.

00:03:53.130 --> 00:03:55.004
So if I were to turn
this parallelogram over,

00:03:55.004 --> 00:03:57.050
it would look
something like this.

00:03:57.050 --> 00:04:00.420
It would look
something like this.

00:04:00.420 --> 00:04:03.970
So if I were to
rotate it like that,

00:04:03.970 --> 00:04:06.750
and stand it on this side,
so this would be point--

00:04:06.750 --> 00:04:10.027
let me draw the points--
this would be point A.

00:04:10.027 --> 00:04:11.610
Let me make sure I'm
doing this right.

00:04:11.610 --> 00:04:11.800
Yeah.

00:04:11.800 --> 00:04:13.175
This would be
point A. This would

00:04:13.175 --> 00:04:15.490
be point D. This
would be point C.

00:04:15.490 --> 00:04:18.990
And then this would be point B.
You could also do it this way.

00:04:18.990 --> 00:04:20.890
You could say it's
1-- sorry, not 1/2.

00:04:20.890 --> 00:04:22.130
That would be for a triangle.

00:04:22.130 --> 00:04:24.110
The area of this would
be base times height.

00:04:24.110 --> 00:04:27.710
So you could say
it's h times DC.

00:04:27.710 --> 00:04:29.900
So you could say
this is going to be

00:04:29.900 --> 00:04:34.991
equal to h times the length DC.

00:04:34.991 --> 00:04:35.990
That's one way to do it.

00:04:35.990 --> 00:04:37.760
That's this base
times this height.

00:04:37.760 --> 00:04:46.520
Or you could say it's
equal to AD times,

00:04:46.520 --> 00:04:51.620
I'll call this altitude right
here, I'll call this height 2.

00:04:51.620 --> 00:04:53.550
Times h2.

00:04:53.550 --> 00:04:55.330
Maybe I'll call this h1.

00:04:55.330 --> 00:04:56.330
h1, h2.

00:04:56.330 --> 00:04:59.450
So you could take this
base times this height,

00:04:59.450 --> 00:05:04.910
or you could take
this base times

00:05:04.910 --> 00:05:06.210
this height right over here.

00:05:06.210 --> 00:05:07.590
This is h2.

00:05:07.590 --> 00:05:08.474
Either way.

00:05:08.474 --> 00:05:10.640
So if someone were to give
you a parallelogram, just

00:05:10.640 --> 00:05:13.580
to make things clear,
obviously, you'd

00:05:13.580 --> 00:05:16.530
have to be have some way to be
able to figure out the height.

00:05:16.530 --> 00:05:18.260
So if someone were to give
you a parallelogram like this,

00:05:18.260 --> 00:05:20.145
they would tell you
this is a parallelogram.

00:05:20.145 --> 00:05:22.520
If they were to tell you that
this length right over here

00:05:22.520 --> 00:05:28.090
is 5, and if they were to tell
you that this distance is 6,

00:05:28.090 --> 00:05:29.600
then the area of
this parallelogram

00:05:29.600 --> 00:05:31.339
would literally be 5 times 6.

00:05:31.339 --> 00:05:33.380
I drew the altitude outside
of the parallelogram.

00:05:33.380 --> 00:05:35.296
I could have drawn it
right over here as well.

00:05:35.296 --> 00:05:36.670
That would also be 6.

00:05:36.670 --> 00:05:41.360
So the area of this
parallelogram would be 30.


Converting repeating decimals to fractions 1

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.300
In this video, I want
to talk about how

00:00:02.300 --> 00:00:05.860
we can convert repeating
decimals into fractions.

00:00:05.860 --> 00:00:08.360
So let's give ourselves
a repeating decimal.

00:00:08.360 --> 00:00:11.390
So let's say I had the
repeating decimal 0.7.

00:00:11.390 --> 00:00:13.250
And sometimes it'll
be written like that,

00:00:13.250 --> 00:00:15.820
which just means that
the 7 keeps on repeating.

00:00:15.820 --> 00:00:21.040
So this is the same
thing as 0.7777

00:00:21.040 --> 00:00:24.020
and I could just keep going
on and on and on forever

00:00:24.020 --> 00:00:25.930
with those 7s.

00:00:25.930 --> 00:00:30.340
So the trick to converting
these things into fractions

00:00:30.340 --> 00:00:33.349
is to essentially set
this equal to a variable.

00:00:33.349 --> 00:00:35.140
And we'll just show
it, do it step-by-step.

00:00:35.140 --> 00:00:36.764
So let me set this
equal to a variable.

00:00:36.764 --> 00:00:38.380
Let me call this x.

00:00:38.380 --> 00:00:42.710
So x is equal to
0.7, and then the 7

00:00:42.710 --> 00:00:44.640
repeats on and on forever.

00:00:44.640 --> 00:00:46.582
Now what would 10x be?

00:00:46.582 --> 00:00:47.790
Well, let's think about this.

00:00:47.790 --> 00:00:48.960
10x.

00:00:48.960 --> 00:00:52.494
10x would just be 10 times this.

00:00:52.494 --> 00:00:54.410
And we could even think
of it right over here.

00:00:54.410 --> 00:00:56.540
It would be, if we
multiplied this times 10,

00:00:56.540 --> 00:00:58.970
you'd be moving the decimal
1 over to the right,

00:00:58.970 --> 00:01:04.110
it would be 7.777, on and
on and on and on forever.

00:01:04.110 --> 00:01:08.800
Or you could say it
is 7.7 repeating.

00:01:08.800 --> 00:01:10.040
Now this is the trick here.

00:01:10.040 --> 00:01:12.550
So let me make these
equal to each other.

00:01:12.550 --> 00:01:16.880
So we know what x is. x is this,
just 0.777 repeating forever.

00:01:16.880 --> 00:01:18.270
10x is this.

00:01:18.270 --> 00:01:20.100
And this is another
repeating thing.

00:01:20.100 --> 00:01:22.530
Now the way that we can get
rid of the repeating decimals

00:01:22.530 --> 00:01:25.400
is if we subtract x from 10x.

00:01:25.400 --> 00:01:25.900
Right?

00:01:25.900 --> 00:01:28.790
Because x has all these 0.7777.

00:01:28.790 --> 00:01:32.102
If you subtract
that from 7.77777,

00:01:32.102 --> 00:01:33.810
then you're just going
to be left with 7.

00:01:33.810 --> 00:01:34.920
So let's do that.

00:01:34.920 --> 00:01:38.220
So let me rewrite it here just
so it's a little bit neater.

00:01:38.220 --> 00:01:42.080
10x is equal to 7.7
repeating, which

00:01:42.080 --> 00:01:46.490
is equal to 7.777
on and on forever.

00:01:46.490 --> 00:01:49.360
And we established
earlier that x

00:01:49.360 --> 00:01:55.900
is equal to 0.7 repeating,
which is equal to 0.777

00:01:55.900 --> 00:01:57.520
on and on and on forever.

00:01:57.520 --> 00:02:00.440
Now what happens if you
subtract x from 10x?

00:02:00.440 --> 00:02:03.400
So we're going to subtract
the yellow from the green.

00:02:03.400 --> 00:02:05.770
Well, 10 of something
minus 1 of something

00:02:05.770 --> 00:02:08.680
is just going to be
9 of that something.

00:02:08.680 --> 00:02:10.650
And then that's
going to be equal to,

00:02:10.650 --> 00:02:16.110
what's 7.7777
repeating minus 0.77777

00:02:16.110 --> 00:02:17.630
going on and on
forever repeating?

00:02:17.630 --> 00:02:19.250
Well it's just going to be 7.

00:02:19.250 --> 00:02:20.750
These parts are
going to cancel out.

00:02:20.750 --> 00:02:21.940
You're just left with 7.

00:02:21.940 --> 00:02:23.980
Or you could say these
two parts cancel out.

00:02:23.980 --> 00:02:26.530
You're just left with 7.

00:02:26.530 --> 00:02:28.800
And so you get 9x is equal to 7.

00:02:28.800 --> 00:02:32.330
To solve for x, you just
divide both sides by 9.

00:02:32.330 --> 00:02:34.882
Let's divide both sides by nine.

00:02:34.882 --> 00:02:37.090
I could do all three sides,
although these are really

00:02:37.090 --> 00:02:38.280
saying the same thing.

00:02:38.280 --> 00:02:43.200
And you get x is equal to 7/9.

00:02:43.200 --> 00:02:45.060
Let's do another one.

00:02:45.060 --> 00:02:47.750
I'll leave this one here
so you can refer to it.

00:02:47.750 --> 00:02:55.210
So let's say I have the
number 1.2 repeating.

00:02:55.210 --> 00:02:59.330
So this is the same thing
as 1.2222 on and on and on.

00:02:59.330 --> 00:03:01.200
Whatever the bar is
on top of, that's

00:03:01.200 --> 00:03:03.700
the part that repeats
on and on forever.

00:03:03.700 --> 00:03:07.520
So just like we did over here,
let's set this equal to x.

00:03:07.520 --> 00:03:09.610
And then let's say 10x.

00:03:09.610 --> 00:03:11.930
Let's multiply this by 10.

00:03:11.930 --> 00:03:17.620
So 10x is equal to, it
would be 12.2 repeating,

00:03:17.620 --> 00:03:22.420
which is the same thing as
12.222 on and on and on and on.

00:03:22.420 --> 00:03:24.805
And then we can
subtract x from 10x.

00:03:24.805 --> 00:03:26.180
And you don't have
to rewrite it,

00:03:26.180 --> 00:03:28.471
but I'll rewrite it here just
so we don't get confused.

00:03:28.471 --> 00:03:32.040
So we have x is equal
to 1.2 repeating.

00:03:32.040 --> 00:03:35.240
And so if we subtract x
from 10x, what do we get?

00:03:35.240 --> 00:03:41.150
On the left-hand side,
we get 10x minus x is 9x.

00:03:41.150 --> 00:03:44.110
And this is going to be
equal to, well, the 2

00:03:44.110 --> 00:03:45.750
repeating parts cancel out.

00:03:45.750 --> 00:03:46.910
This cancels with that.

00:03:46.910 --> 00:03:48.660
If 2 repeating
minus 2 repeating,

00:03:48.660 --> 00:03:49.780
that's just a bunch of 0.

00:03:49.780 --> 00:03:52.810
So it's 12 minus 1 is 11.

00:03:52.810 --> 00:03:54.720
And you have 9x is equal to 11.

00:03:54.720 --> 00:03:57.060
Divide both sides by 9.

00:03:57.060 --> 00:04:01.635
You get x is equal to 11/9.


Proof: Rhombus area half product of diagonal length

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.594
So quadrilateral ABCD, they're
telling us it is a rhombus,

00:00:04.594 --> 00:00:06.260
and what we need to
do, we need to prove

00:00:06.260 --> 00:00:12.120
that the area of this rhombus is
equal to 1/2 times AC times BD.

00:00:12.120 --> 00:00:15.040
So we're essentially proving
that the area of a rhombus

00:00:15.040 --> 00:00:18.970
is 1/2 times the product of
the lengths of its diagonals.

00:00:18.970 --> 00:00:21.020
So let's see what
we can do over here.

00:00:21.020 --> 00:00:23.800
So there's a bunch of
things we know about rhombi

00:00:23.800 --> 00:00:26.370
and all rhombi are
parallelograms,

00:00:26.370 --> 00:00:28.920
so there's tons of things that
we know about parallelograms.

00:00:28.920 --> 00:00:31.980
First of all, if it's a rhombus,
we know that all of the sides

00:00:31.980 --> 00:00:33.080
are congruent.

00:00:33.080 --> 00:00:35.390
So that side length is
equal to that side length is

00:00:35.390 --> 00:00:38.460
equal to that side length is
equal to that side length.

00:00:38.460 --> 00:00:40.500
Because it's a
parallelogram, we know

00:00:40.500 --> 00:00:42.530
the diagonals bisect each other.

00:00:42.530 --> 00:00:44.145
So we know that
this length-- let

00:00:44.145 --> 00:00:46.370
me call this point
over here B, let's

00:00:46.370 --> 00:00:50.130
call this E. We know that BE
is going to be equal to ED.

00:00:50.130 --> 00:00:53.370
So that's BE, we know that's
going to be equal to ED.

00:00:53.370 --> 00:00:58.570
And we know that
AE is equal to EC.

00:00:58.570 --> 00:01:00.410
We also know, because
this is a rhombus,

00:01:00.410 --> 00:01:02.579
and we proved this
in the last video,

00:01:02.579 --> 00:01:04.870
that the diagonals, not only
do they bisect each other,

00:01:04.870 --> 00:01:06.840
but they are also perpendicular.

00:01:06.840 --> 00:01:08.910
So we know that this
is a right angle,

00:01:08.910 --> 00:01:11.660
this is a right angle,
that is a right angle,

00:01:11.660 --> 00:01:13.760
and then this is a right angle.

00:01:13.760 --> 00:01:15.860
So the easiest way
to think about it

00:01:15.860 --> 00:01:19.690
is if we can show that
this triangle ADC is

00:01:19.690 --> 00:01:23.230
congruent to triangle ABC,
and if we can figure out

00:01:23.230 --> 00:01:25.480
the area of one of them,
we can just double it.

00:01:25.480 --> 00:01:28.000
So the first part is
pretty straightforward.

00:01:28.000 --> 00:01:35.430
So we can see that
triangle ADC is

00:01:35.430 --> 00:01:42.300
going to be congruent
to triangle ABC,

00:01:42.300 --> 00:01:45.720
and we know that by
side-side-side congruency.

00:01:45.720 --> 00:01:47.640
This side is congruent
to that side.

00:01:47.640 --> 00:01:50.320
This side is congruent to
that side, and they both share

00:01:50.320 --> 00:01:51.940
a C right over here.

00:01:51.940 --> 00:01:54.630
So this is by side-side-side.

00:01:54.630 --> 00:01:59.640
And so we can say that the
area-- so because of that,

00:01:59.640 --> 00:02:04.400
we know that the
area of ABCD is just

00:02:04.400 --> 00:02:09.304
going to be equal to
2 times the area of,

00:02:09.304 --> 00:02:10.720
we could pick
either one of these.

00:02:10.720 --> 00:02:12.270
We could say 2 times
the area of ABC.

00:02:16.450 --> 00:02:19.110
Because area of ABCD-- actually
let me write it this way.

00:02:19.110 --> 00:02:26.100
The area of ABCD is equal to
the area of ADC plus the area

00:02:26.100 --> 00:02:27.687
of ABC.

00:02:27.687 --> 00:02:29.270
But since they're
congruent, these two

00:02:29.270 --> 00:02:31.103
are going to be the
same thing, so it's just

00:02:31.103 --> 00:02:35.520
going to be 2 times
the area of ABC.

00:02:35.520 --> 00:02:39.190
Now what is the area of ABC?

00:02:39.190 --> 00:02:43.200
Well area of a triangle is
just 1/2 base times height.

00:02:43.200 --> 00:02:51.010
So area of ABC is just
equal to 1/2 times the base

00:02:51.010 --> 00:02:54.200
of that triangle times its
height, which is equal to 1/2.

00:02:54.200 --> 00:02:56.040
What is the length of the base?

00:02:56.040 --> 00:02:57.670
Well the length
of the base is AC.

00:02:57.670 --> 00:02:59.670
So it's 1/2-- I'll
color code it.

00:02:59.670 --> 00:03:02.470
The base is AC.

00:03:02.470 --> 00:03:04.310
And then what is the height?

00:03:04.310 --> 00:03:05.910
What is the height here?

00:03:05.910 --> 00:03:09.400
Well we know that this
diagonal right over here,

00:03:09.400 --> 00:03:11.470
that it's a
perpendicular bisector.

00:03:11.470 --> 00:03:15.750
So the height is just
the distance from BE.

00:03:15.750 --> 00:03:22.350
So it's AC times BE,
that is the height.

00:03:22.350 --> 00:03:24.320
This is an altitude.

00:03:24.320 --> 00:03:27.500
It intersects this base
at a 90-degree angle.

00:03:27.500 --> 00:03:32.590
Or we could say BE is the
same thing as 1/2 times BD.

00:03:32.590 --> 00:03:34.220
So this is-- let me write it.

00:03:34.220 --> 00:03:41.080
This is equal to, so it's
equal to 1/2 times AC, that's

00:03:41.080 --> 00:03:42.060
our base.

00:03:42.060 --> 00:03:44.100
And then our height is
BE, which we're saying

00:03:44.100 --> 00:03:46.620
is the same thing
as 1/2 times BD.

00:03:51.470 --> 00:03:54.000
So that's the area
of just ABC, that's

00:03:54.000 --> 00:03:56.506
just the area of this broader
triangle right up there,

00:03:56.506 --> 00:03:58.130
or that larger triangle
right up there,

00:03:58.130 --> 00:03:59.332
that half of the rhombus.

00:03:59.332 --> 00:04:01.290
But we decided that the
area of the whole thing

00:04:01.290 --> 00:04:03.920
is two times that.

00:04:03.920 --> 00:04:06.280
So if we go back, if we
use both this information

00:04:06.280 --> 00:04:12.220
and this information right over
here, we have the area of ABCD

00:04:12.220 --> 00:04:15.460
is going to be equal to
2 times the area of ABC,

00:04:15.460 --> 00:04:21.269
where the area of ABC is
this thing right over here.

00:04:21.269 --> 00:04:24.459
So 2 times the area
of ABC, area of ABC

00:04:24.459 --> 00:04:25.500
is that right over there.

00:04:25.500 --> 00:04:31.709
So 1/2 times 1/2 is
1/4 times AC times BD.

00:04:31.709 --> 00:04:33.250
And then you see
where this is going.

00:04:33.250 --> 00:04:38.760
2 times 1/4 fourth is
1/2 times AC times BD.

00:04:38.760 --> 00:04:40.826
Fairly straightforward,
which is a neat result.

00:04:40.826 --> 00:04:42.700
And actually, I haven't
done this in a video.

00:04:42.700 --> 00:04:43.880
I'll do it in the next video.

00:04:43.880 --> 00:04:46.290
There are other ways of finding
the areas of parallelograms,

00:04:46.290 --> 00:04:46.789
generally.

00:04:46.789 --> 00:04:50.710
It's essentially base times
height, but for a rhombus

00:04:50.710 --> 00:04:52.780
we could do that because
it is a parallelogram,

00:04:52.780 --> 00:04:55.050
but we also have this
other neat little result

00:04:55.050 --> 00:04:56.716
that we proved in this video.

00:04:56.716 --> 00:04:58.590
That if we know the
lengths of the diagonals,

00:04:58.590 --> 00:05:02.800
the area of the rhombus
is 1/2 times the products

00:05:02.800 --> 00:05:06.480
of the lengths of the diagonals,
which is kind of a neat result.


Proof: Rhombus diagonals are perpendicular bisectors

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.465
We're told that quadrilateral
ABCD is a rhombus.

00:00:04.465 --> 00:00:07.670
And what they want us to
prove is that their diagonals

00:00:07.670 --> 00:00:11.340
are perpendicular, that
AC is perpendicular to BD.

00:00:11.340 --> 00:00:14.100
Now let's think about everything
we know about a rhombus.

00:00:14.100 --> 00:00:16.379
First of all, a rhombus
is a special case

00:00:16.379 --> 00:00:17.690
of a parallelogram.

00:00:17.690 --> 00:00:20.620
In a parallelogram, the
opposite sides are parallel.

00:00:20.620 --> 00:00:22.630
So that side is
parallel to that side.

00:00:22.630 --> 00:00:24.890
These two sides are parallel.

00:00:24.890 --> 00:00:28.330
And in a rhombus, not only are
the opposite sides parallel--

00:00:28.330 --> 00:00:30.670
it's a parallelogram--
but also, all of the sides

00:00:30.670 --> 00:00:31.920
have equal length.

00:00:31.920 --> 00:00:35.060
So this side is equal
to this side, which

00:00:35.060 --> 00:00:38.170
is equal to that side,
which is equal to that side

00:00:38.170 --> 00:00:39.200
right over there.

00:00:39.200 --> 00:00:40.860
Now, there's other
interesting things

00:00:40.860 --> 00:00:43.740
we know about the diagonals
of a parallelogram, which

00:00:43.740 --> 00:00:46.040
we know all rhombi
are parallelograms.

00:00:46.040 --> 00:00:48.840
The other way around is
not necessarily true.

00:00:48.840 --> 00:00:51.670
We know that for any
parallelogram-- and a rhombus

00:00:51.670 --> 00:00:55.660
is a parallelogram-- that the
diagonals bisect each other.

00:00:55.660 --> 00:00:58.210
So for example, let me label
this point in the center.

00:00:58.210 --> 00:00:59.885
Let me label it point E.

00:00:59.885 --> 00:01:04.040
We know that AE is
going to be equal to EC.

00:01:04.040 --> 00:01:06.560
I'll put two slashes
right over there.

00:01:06.560 --> 00:01:13.290
And we also know that EB
is going to be equal to ED.

00:01:13.290 --> 00:01:16.820
So this is all of what
we know when someone just

00:01:16.820 --> 00:01:20.350
says that ABCD is a rhombus,
based on other things

00:01:20.350 --> 00:01:22.300
that we've proven to ourselves.

00:01:22.300 --> 00:01:27.000
Now we need to prove that
AC is perpendicular to BD.

00:01:27.000 --> 00:01:29.550
So an interesting way to prove
it-- and you can look at it

00:01:29.550 --> 00:01:32.770
just by eyeballing it-- is if we
can show that this triangle is

00:01:32.770 --> 00:01:36.592
congruent to this triangle and
that these two angles right

00:01:36.592 --> 00:01:38.050
over here correspond
to each other,

00:01:38.050 --> 00:01:39.290
then they have to be the same.

00:01:39.290 --> 00:01:41.831
And they'll be supplementary,
and then they'll be 90 degrees.

00:01:41.831 --> 00:01:43.530
So let's just prove
it to ourselves.

00:01:43.530 --> 00:01:47.180
So the first thing we see
is we have a side, a side,

00:01:47.180 --> 00:01:50.430
and a side; a side,
a side, and a side.

00:01:50.430 --> 00:01:54.108
So we can see that triangle--
let me write it here.

00:01:54.108 --> 00:01:56.573
Let me [? do this ?]
in a new color.

00:01:56.573 --> 00:02:08.655
We see that triangle ABE is
congruent to triangle CBE.

00:02:11.649 --> 00:02:15.930
And we know that by
side-side-side congruency.

00:02:15.930 --> 00:02:18.969
We have a side, a side,
and a side; a side, a side,

00:02:18.969 --> 00:02:20.402
and a side.

00:02:20.402 --> 00:02:21.860
And then once we
know that, we know

00:02:21.860 --> 00:02:24.120
that all the corresponding
angles are congruent.

00:02:24.120 --> 00:02:31.380
And in particular, we
know that angle AEB

00:02:31.380 --> 00:02:35.948
is going to be
congruent to angle CEB.

00:02:39.860 --> 00:02:48.810
Because they are corresponding
angles of congruent triangles.

00:02:48.810 --> 00:02:51.070
So this angle right
over here is going

00:02:51.070 --> 00:02:53.380
to be equal to that
angle over there.

00:02:53.380 --> 00:02:55.456
And we also know that
they are supplementary.

00:03:01.676 --> 00:03:02.800
Let me write it this way.

00:03:02.800 --> 00:03:08.550
They're congruent, and
they are supplementary.

00:03:08.550 --> 00:03:10.950
These two are going to
have the same measure,

00:03:10.950 --> 00:03:13.480
and they need to add
up to 180 degrees.

00:03:13.480 --> 00:03:15.470
So if I have two things
that are the same thing

00:03:15.470 --> 00:03:18.340
and they add up to 180 degrees,
what does that tell me?

00:03:18.340 --> 00:03:23.850
So that tells me that
the measure of angle AEB

00:03:23.850 --> 00:03:29.140
is equal to the measure
of angle CEB, which

00:03:29.140 --> 00:03:31.730
must be equal to 90 degrees.

00:03:31.730 --> 00:03:34.910
They're the same measure,
and they are supplementary.

00:03:34.910 --> 00:03:37.710
So this is a right angle, and
then this is a right angle.

00:03:37.710 --> 00:03:39.380
And obviously, if
this is a right angle,

00:03:39.380 --> 00:03:40.890
this angle down here
is a vertical angle.

00:03:40.890 --> 00:03:42.080
That's going to
be a right angle.

00:03:42.080 --> 00:03:43.550
If this is a right
angle, this over here

00:03:43.550 --> 00:03:44.883
is going to be a vertical angle.

00:03:44.883 --> 00:03:48.640
And you see the diagonals
intersect at a 90-degree angle.

00:03:48.640 --> 00:03:51.120
So we've just proved--
so this is interesting.

00:03:51.120 --> 00:03:54.120
A parallelogram, the
diagonals bisect each other.

00:03:54.120 --> 00:03:57.350
For a rhombus, where
all the sides are equal,

00:03:57.350 --> 00:03:59.610
we've shown that not only
do they bisect each other

00:03:59.610 --> 00:04:03.840
but they're perpendicular
bisectors of each other.