What if you were told that a triangle had side lengths of 5, 12, and 13? How could you determine if the triangle were a right one? After completing this Concept, you'll be able to use the Pythagorean Theorem to solve problems like this one.

### Watch This

CK-12 Foundation: Using The Pythagorean Theorem

James Sousa: Pythagorean Theorem

### Guidance

The two shorter sides of a right triangle (the sides that form the right angle) are the
**
legs
**
and the longer side (the side opposite the right angle) is the
**
hypotenuse
**
. For the Pythagorean Theorem, the legs are “
” and “
” and the hypotenuse is “
”.

**
Pythagorean Theorem:
**
Given a right triangle with legs of lengths
and
and a hypotenuse of length
,
.

For proofs of the Pythagorean Theorem go to: http://www.mathsisfun.com/pythagoras.html and scroll down to “And You Can Prove the Theorem Yourself.”

The converse of the Pythagorean Theorem is also true. It allows you to prove that a triangle is a right triangle even if you do not know its angle measures.

**
Pythagorean Theorem Converse:
**
If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If , then is a right triangle.

##### Pythagorean Triples

A combination of three numbers that makes the Pythagorean Theorem true is called a
**
Pythagorean triple.
**
Each set of numbers below is a Pythagorean triple.

Any multiple of a Pythagorean triple is also considered a Pythagorean triple. Multiplying 3, 4, 5 by 2 gives 6, 8, 10, which is another triple. To see if a set of numbers makes a Pythagorean triple, plug them into the Pythagorean Theorem.

#### Example A

Do 6, 7, and 8 make the sides of a right triangle?

Plug the three numbers into the Pythagorean Theorem. Remember that the largest length will always be the hypotenuse, . If , then they are the sides of a right triangle.

#### Example B

Find the length of the hypotenuse.

Use the Pythagorean Theorem. Set and . Solve for .

#### Example C

Is 20, 21, 29 a Pythagorean triple?

If , then the set is a Pythagorean triple.

Therefore, 20, 21, and 29 is a Pythagorean triple.

#### Example D

Determine if the triangles below are right triangles.

a)

b)

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent .

a)

b)

CK-12 Foundation: Using The Pythagorean Theorem

### Guided Practice

1. Find the missing side of the right triangle below.

2. What is the diagonal of a rectangle with sides 10 and 16?

3. Do the following lengths make a right triangle?

a)

b)

c)

**
Answers:
**

1. Here, we are given the hypotenuse and a leg. Let’s solve for .

2. For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find .

3. Even though there is no picture, you can still use the Pythagorean Theorem. Again, the longest length will be .

a)

b)

c) This is a multiple of of a 3, 4, 5 right triangle. Yes, this is a right triangle.

### Practice

Find the length of the missing side. Simplify all radicals.

- If the legs of a right triangle are 10 and 24, then the hypotenuse is __________.
- If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
- If the sides of a square are 16, then the diagonal is ____________.
- If the sides of a square are 9, then the diagonal is _____________.

Determine if the following sets of numbers are Pythagorean Triples.

- 12, 35, 37
- 9, 17, 18
- 10, 15, 21
- 11, 60, 61
- 15, 20, 25
- 18, 73, 75

Determine if the following lengths make a right triangle.

- 7, 24, 25
- 15, 20, 25
- 20, 25, 30