1. Converse of the Pythagorean Theorem
Lesson 2.4
Core Focus on Geometry
Converse of the Pythagorean Theorem

2. Warm-Up
Solve for x 4 5 7 24 12 16 x x x
x = 25
x = 20
x = 3

3. Converse of the Pythagorean Theorem
Lesson 2.4
Converse of the Pythagorean Theorem
Determine if three side lengths create a right triangle.

4. The Converse of the Pythagorean Theorem
If a2 + b2 = c2, then the triangle is a right triangle. a b c

5. Explore! Make It Right
Cory and Paco each chose three side lengths they believe will form a right triangle. Cory chose the lengths 8, 10 and 18. Paco chose 5, 12 and 13.
Step 1 Begin with Cory’s set of numbers: 8, 10 and 18.
a. Which measure is the hypotenuse? How do you know?
b. Which measures represent the legs of the triangle?
c. Substitute the values into the Pythagorean Theorem for a, b and c. Simplify the equation. Does one side of the equation equal the other side of the equation? If so, it is a right triangle.
a2 + b2 = c2
Step 2 Repeat the process in Step 1 using Paco’s set of numbers: 5, 12 and 13.
Do Paco’s side lengths form a right triangle?
Step 3 Find at least one other set of measures that will form a right triangle.

6. Good to Know!
Three positive integers that work in the Pythagorean Theorem are called Pythagorean triples. Recognizing the common Pythagorean triples will save you time when you find them in problems or real-world situations. Some common sets of Pythagorean triples are: Notice that the Pythagorean triples in the second row are multiples of a Pythagorean triple in the top row. You can create an infinite number of Pythagorean triples by multiplying all numbers in a Pythagorean triple by a constant.
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
9, 12, 15
10, 24, 26
6, 8, 10

7. Pythagorean Triples a2 + b2 = c2
A Pythagorean triple is a set of three positive integers such that
a2 + b2 = c2

8. It does not matter which leg is a and which leg is b.
Example 1
It does not matter which leg is a and which leg is b.
A triangle has side lengths of 4, 10 and 9 inches. Determine if this triangle is a right triangle. The largest measure is the hypotenuse. c = 10 The other two measures are the legs. a = 4 b = 9 Write the Pythagorean Theorem. a2 + b2 = c2 Substitute the given values for = 102 the hypotenuse and legs. Simplify by squaring = 100 Check to see if one side of the 97 100 equation equals the other side. A triangle with side lengths of 4, 10 and 9 is not a right triangle.

9. Communication Prompts
Do you think it would be helpful to have some of the common Pythagorean Triples memorized? Why?
Right angles are important in every day life. List at least 5 different ways right angles are used in the world today.

10. Exit Problems
Determine if each triangle is a right triangle. If it is, state if its side lengths form a Pythagorean Triple 4 6 6 10 6
6.5 8
2.5 8
Yes;
Pythagorean triple
Yes;
not a Pythagorean triple No