Main content

## High school geometry

# Special right triangles proof (part 2)

CCSS.Math:

Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2). Created by Sal Khan.

## Video transcript

In the last video, we
showed that the ratios of the sides of a
30-60-90 triangle are-- if we assume
the longest side is x, if the hypotenuse is x. Then the shortest side is
x/2 and the side in between, the side that's opposite
the 60 degree side, is square root of 3x/2. Or another way to think about it
is if the shortest side is 1-- Now I'll do the shortest side,
then the medium size, then the longest side. So if the side opposite
the 30 degree side is 1, then the side opposite
the 60 degree side is square root of 3 times that. So it's going to be
square root of 3. And then the hypotenuse
is going to be twice that. In the last video,
we started with x and we said that the
30 degree side is x/2. But if the 30 degree
side is 1, then this is going to be twice that. So it's going to be 2. This right here is the side
opposite the 30 degree side, opposite the 60 degree side,
and then the hypotenuse opposite the 90 degree side. And so, in general, if
you see any triangle that has those ratios, you say hey,
that's a 30-60-90 triangle. Or if you see a
triangle that you know is a 30-60-90 triangle,
you could say, hey, I know how to figure out
one of the sides based on this ratio right over here. Just an example, if
you see a triangle that looks like this, where the
sides are 2, 2 square root of 3, and 4. Once again, the ratio of
2 to 2 square root of 3 is 1 to square root of 3. The ratio of 2 to 4 is
the same thing as 1 to 2. This right here must
be a 30-60-90 triangle. What I want to introduce
you to in this video is another important
type of triangle that shows up a lot in geometry
and a lot in trigonometry. And this is a 45-45-90 triangle. Or another way to
think about is if I have a right triangle
that is also isosceles. You obviously can't have a right
triangle that is equilateral, because an equilateral triangle
has all of their angles have to be 60 degrees. But you can have
a right angle, you can have a right triangle,
that is isosceles. And isosceles--
let me write this-- this is a right
isosceles triangle. And if it's isosceles,
that means two of the sides are equal. So these are the two
sides that are equal. And then if the two
sides are equal, we have proved to ourselves
that the base angles are equal. And if we called the measure
of these base angles x, then we know that x plus x plus
90 have to be equal to 180. Or if we subtract
90 from both sides, you get x plus x is equal
to 90 or 2x is equal to 90. Or if you divide
both sides by 2, you get x is equal
to 45 degrees. So a right isosceles
triangle can also be called-- and this is the more
typical name for it-- it can also be called
a 45-45-90 triangle. And what I want to do
this video is come up with the ratios for the
sides of a 45-45-90 triangle, just like we did for
a 30-60-90 triangle. And this one's actually
more straightforward. Because in a 45-45-90 triangle,
if we call one of the legs x, the other leg is
also going to be x. And then we can use
the Pythagorean Theorem to figure out the length
of the hypotenuse. So the length of the
hypotenuse, let's call that c. So we get x squared
plus x squared. That's the square of
length of both of the legs. So when we sum those
up, that's going to have to be
equal to c squared. This is just straight out
of the Pythagorean theorem. So we get 2x squared
is equal to c squared. We can take the principal
root of both sides of that. I wanted to just
change it to yellow. Last, take the principal
root of both sides of that. The left-hand side you
get, principal root of 2 is just square
root of 2, and then the principal root of x
squared is just going to be x. So you're going to have x
times the square root of 2 is equal to c. So if you have a right isosceles
triangle, whatever the two legs are, they're going
to have the same length. That's why it's isosceles. The hypotenuse is going to be
square root of 2 times that. So c is equal to x times
the square root of 2. So for example, if you have a
triangle that looks like this. Let me draw it a
slightly different way. It's good to have to orient
ourselves in different ways every time. So if we see a triangle
that's 90 degrees, 45 and 45 like this,
and you really just have to know two of
these angles to know what the other one
is going to be, and if I tell you that
this side right over here is 3-- I actually don't
even have to tell you that this other
side's going to be 3. This is an isosceles
triangle, so those two legs are going to be the same. And you won't even have to
apply the Pythagorean theorem if you know this--
and this is a good one to know-- that the hypotenuse
here, the side opposite the 90 degree side, is just going
to be square root of 2 times the length of
either of the legs. So it's going to be 3
times the square root of 2. So the ratio of the
size of the hypotenuse in a 45-45-90 triangle or
a right isosceles triangle, the ratio of the sides are
one of the legs can be 1. Then the other leg is going
to have the same measure, the same length, and then
the hypotenuse is going to be square root of 2
times either of those. 1 to 1, 2 square root of 2. So this is 45-45-90. That's the ratios. And just as a review,
if you have a 30-60-90, the ratios were 1 to
square root of 3 to 2. And now we'll apply this
in a bunch of problems.