Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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A 45 45 90 triangle is a special type of isosceles right triangle where the two legs are congruent to one another and the non-right angles are both equal to 45 degrees. Many times, we can use the Pythagorean theorem to find the missing legs or hypotenuse of **45 45 90 triangles**. The ratio of the sides to the hypotenuse is always 1:1:square root of two.

Something special in geometry

is the 45, 45, 90 triangle.

Well, a 45, 45, 90 triangle is an isosceles

right triangle where these two legs

are congruent to each other.

The reason why it's 45, 45, 90 is because

if we know that these two angles are

congruent to each other, because the

isosceles triangle theorem, then

we can say that 180 degrees is equal

to 90, plus X plus X. So if

I add these up, I'm going to have 180

is equal to 90, plus 2 X, so I'm

going to subtract 90 from both sides

and I get 90 is equal to 2X, and

then I'm going to divide by 2 to

solve for X. And 90 divided by

2 is 45, which means each of these angles

that are congruent to each other

have to be 45 degrees.

So in an isosceles right triangle you're

going to have a 45 degree, a 45 degree

and a 90 degree.

So that's we mean when

we say 45, 45, 90.

Now something is going on with

these angles and sides.

And if I wrote in that these were both X and

I would say that this is my hypotenuse

C, let's apply the Pythagorean

theorem and see what happens.

Pythagorean theorem says A squared plus

B squared equals C squared and A and

B here are both X. So I'm going to

write that X squared plus X squared

is equal to C squared.

I can combine like terms here and X squared

plus X squared is 2X squared.

So if I want to solve for my hypotenuse

C, I'm going to take the square root

of both sides, and the square root

of X squared is X, and there is no

whole number square root of 2. So

C is equal to X times the square

root of 2. Well, that's a little

difficult to understand.

So let's say we had an isosceles right triangle

with sides of length 1 and I'm

trying to find the hypotenuse.

So maybe this will make sense

with this triangle.

Here we'll have 1 squared plus 1

squared is equal to C squared

Well, 1 plus 1 is 2. So if I take the

square root of both sides, I find

that my hypotenuse is equal to the

square root of 2. So now what

I see it's talking about is if you know

the side of one of your legs, if

you know that length, you're going to.

multiply it by the square root of 2.

So to get from the leg in a 45, 45, 90.

triangle, you're going to multiply by

the square root of 2.

Let's say, however, you don't know what that

leg is. And you know the hypotenuse.

So I'm going to draw another

triangle over here.

45, 45, 90, and let's say you said this

was 3. To go from your hypotenuse

to your leg, you're going to undo multiplying

by the square root of 2.

So you're going to divide by the square

root of 2. So this answer

right here will be 3 divided by the

square root of 2.

And we can't

have a square root in our denominator here.

So now this is becoming quite a chore.

We're going to multiply by square root

of 2. Multiply by the square root

of 2. So we'll have in our numerator

3 times the square root of

2. Square root of 2 times square root of 2

is 2 because you'll have the square root

of 4. So this is actually 3 times.

the square root of 2 divided by 2.

So if we go back to our original

drawing here where we said.

for any right triangle where you have

two legs that are congruent, to go

from your leg to your hypotenuse, all

you need to do is take that number

and multiply it by the square root of

2. So if X is 5, your hypotenuse

is 5 times the square

root of 2. To go from your hypotenuse back to one of

your legs, you're going to divide by

the square root of 2.

So keep that in mind and solving for missing

sides, an isosceles right triangle

is pretty simple.