45 45 90 Triangles

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.