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.