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## Geometry (all content)

### Unit 7: Lesson 9

Advanced area with triangles# Area of diagonal-generated triangles

Watch Sal prove that the areas of the pairs of triangles generated by the diagonals of a rectangle are equal. Created by Sal Khan.

## Video transcript

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