This lesson will teach you about one of the special right triangles, the 45-45-90 triangle. You will learn the formulas for calculating the lengths of the sides of this type of triangle. After the lesson, you will be able to test your knowledge with a quiz.

## Rules of a 45-45-90 Triangle

When we are talking about a **45-45-90 triangle**, those numbers represent the measures of the angles of that triangle. So, it means the triangle has two 45-degree angles and one 90-degree angle.Let’s do a little origami to make one of these triangles! Don’t worry; you don’t have to make a swan! Origami is usually made using a square piece of paper.

Imagine you take a square and fold it so that two of the corners end up on top of each other, like this:

We placed corner B on corner D and made a crease diagonally through corners A and C. Now we have made a triangle.

Since we did not cut anything off the paper, we know several things about that triangle. The angle at D is a **right angle** (measures 90 degrees) because the square had four right angles. The angles A and C must each be 45 degrees because the right angles there were folded exactly in half. Since we folded exactly through the corners, the lengths of the sides AD and DC did not change from what they were in the square. Therefore, we know that the sides AD and DC must be of equal length because on the square, all of the sides were of equal length.

Doing this little folding exercise, we have discovered that every 45-45-90 triangle has two sides with the same length. Those sides are called the **legs** of the triangle. The **hypotenuse** (the side opposite of the right angle) will always have a length longer than the legs. In the next two sections, we will talk about the formula to calculate the length of the hypotenuse and the theorem for 45-45-90 triangles.

## Formula and Theorem

The relationship between the three sides of any kind of right triangle is given by the Pythagorean Theorem.

The formula for the **Pythagorean Theorem** is *a*2 + *b*2 = *c*2. The rule for using this formula is that *c* must stand for the hypotenuse. It does not matter which of the two sides you call *a* and which you call *b*.Now, let’s take a look at how the Pythagorean Theorem works for a generic 45-45-90 triangle. That way, we can figure out a formula that relates the length of the hypotenuse to the length of the leg of any 45-45-90 triangle.In triangle FTY, we labeled the hypotenuse *c* according to the rule of the Pythagorean Theorem. The other two sides, as we mentioned above, have the same length, so we do not have to label one of them *a* and the other *b*.

Instead, we can assign both legs the same variable. Let’s use *g*.

Now, let’s fill in the Pythagorean Theorem formula and then do a little algebra to clean things up.*a*2 + *b*2 = *c*2 is the standard Pythagorean Theorem formula. Since we decided to call the two legs by the same name, *g*, in our case we get:*g*2 + *g*2 = *c*2We can simplify this equation by combining the two *g* terms:2*g*2 = *c*2Now let’s solve this equation for *c* by taking the square root of both sides.

Then, we get:*g* * ;2 = *c**c* = *g* * ;2That tells us that for every 45-45-90 triangle, the length of the hypotenuse equals the length of the leg multiplied by square root of 2. That is the **45-45-90 Triangle Theorem**. To think of it another way, if you know the length of the hypotenuse of a 45-45-90 triangle, you can divide the length of the hypotenuse by the square root of 2 to get the length of the leg.Let’s recap the formulas from the 45-45-90 Triangle Theorem here in an easy-to-read form:

## Lesson Summary

**45-45-90 triangle** means a triangle with two 45 degree angles and one 90 degree angle.

A 45-45-90 triangle has two sides that are of equal length, called the **legs**. The third side is longer than the other two and is called the **hypotenuse** and is always opposite the right angle. The relationship between the length of the leg and the length of the hypotenuse is given by the formulas:

## 45-45-90 Triangle Recap

- A 45-45-90 triangle has two 45 degree angles and a right angle
- The two legs of a 45-45-90 triangle are always equal
- The hypotenuse of the triangle is always opposite the right angle
- There are two formulas for the lengths of the sides of a 45-45-90 triangle:

## Learning Outcomes

Once you are finished, you should be able to:

- Describe a 45-45-90 triangle
- Identify the legs and hypotenuse of a triangle
- Recite the Pythagorean Theorem and explain how it relates to the 45-45-90 triangle
- State the 45-45-90 Triangle Theorem and its two formulas