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.
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.
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.
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.
Find the length of the hypotenuse.
Use the Pythagorean Theorem. Set and . Solve for .
Is 20, 21, 29 a Pythagorean triple?
If , then the set is a Pythagorean triple.
Therefore, 20, 21, and 29 is a Pythagorean triple.
Determine if the triangles below are right triangles.
Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent .
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?
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 .
c) This is a multiple of of a 3, 4, 5 right triangle. Yes, this is a right triangle.
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