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Let's say we've got a rectangle
and we have two diagonals
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across the rectangle-- that's
one of them, and then we have
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the other diagonal --and this
rectangle has a height of h--
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so that distance right there is
h --and it has a width of w.
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What we're going to show in
this video is that all of
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these four triangles
have the same area.
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Now right when you look at it,
it might be reasonably obvious
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that this bottom triangle will
have the same area as the top
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triangle, as this top kind
of upside down triangle.
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That these to have the same
area, that might be reasonably
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obvious. they have the same
dimension for their base, this
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width, and they have the same
height because this distance
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right here is exactly half of
the height of the rectangle.
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They are symmetric; they
are equal triangles.
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They have the same proportions.
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Now it's probably equally
obvious that this triangle on
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the left has the same area as
this triangle on the right.
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That's probably
equally obvious.
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What is not obvious is that
these orange triangles angles
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have the same area as these
green, blue triangles.
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And that's what we're
going to show right here.
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So all we have to do is really
calculate the areas of
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the different triangles.
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So let's do the orange
triangles first. and before
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doing that let's just
remind ourselves what the
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area of a triangle is.
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Area of a triangle is equal
to 1/2 times the base of
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the triangle times the
height of the triangle.
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That's just basic geometry.
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Not with that said, let's
figure out the area of
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the orange triangle.
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It's going to be 1/2
times the base.
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So the base of the orange
triangle is this distance
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right here: it is w.
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So 1/2 times w.
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I want to do that in a
different color; the
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color I wrote the w in.
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Now what's the height here?
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Well we already talked about
it: it's exactly half way up
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the height of the rectangle.
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So times 1/2 times the
height of the rectangle.
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So what's that going to be?
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You have 1/2 times 1/2 is 1/4
times width times height.
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So the area of that triangle
is 1/4 width height.
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So is that one.
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Same exact argument;
they have equal area.
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Now what's the area of
these green or these
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green/blue triangles?
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Well once again-- we'll do
this in a green color --area
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is equal to 1/2 base.
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So these guys are
turned on their side.
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The best base I can think of
is this distance right here.
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Or if you look at this triangle
it's this distance right here;
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it is the height of the
rectangle So now we're dealing
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with, the base in this case is
the height of the rectangle.
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Don't want you to
get too confused.
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The height is now
going to be what?
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So these triangles are turned
on the side, so what is
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this distance right here?
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Well it is exactly half
of the width, right?
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We're going exactly half of
this distance right here.
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This point right here is
exactly halfway between
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these two sides and halfway
between those two sides.
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So this distance right
here is 1/2 the width.
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Or the height of these sideways
triangles are 1/2 of the width.
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Little confusing: the base is
equal to the height of the
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rectangle, the height is equal
to 1/2 of the width. but if you
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do the math here, area is equal
to 1/2 times 1/2, which is
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1/4, height times width.
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Or you can just write that as
1/4 width times height, which
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is the exact same area.
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So the area here is 1/4 width
times height, which is the
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exact same area as each of
these orange triangles.
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And it makes sense because
each of them are exactly 1/4
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the area of the rectangle.
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Hopefully you enjoyed that.
