Lesson 9.6 Families of Right Triangles Objective: After studying this lesson you will be able to recognize groups of whole numbers know as Pythagorean triples and apply the principle of the reduced triangle

What are Pythagorean triples? Why should we learn them? They are a combination of whole numbers that satisfy the Pythagorean Theorem. Why should we learn them? Knowing them is not essential but can save you time in doing problems in homework or on a test.

Any three whole numbers that satisfy the equation a2 + b2 = c2 Definition Any three whole numbers that satisfy the equation a2 + b2 = c2 form a Pythagorean Triple Let’s look at some examples 4 3 5 8 6 10 15 9 12

Did you notice any relationships between the triangles? All of the triangles were members of the 3-4-5 family. The 6-8-10 triangle can actually be broken down to The last triangle is NOT a Pythagorean Triple because they are not whole numbers Some common families are (5,12,13), (7, 24, 25), (8,15,17) and there are many more that are not commonly used.

The Principle of the Reduced Triangle Let’s look at trying to simplify some of our work. Example 1 The fraction may complicate our work. So lets multiply value by 2 to get rid of the fractions x 4 7 1/2 By doing this we can clear the fraction and see that we have a Pythagorean Triple (8,15,17). Since 2x = 17, x from our original triangle is 8 ½ .

Principle of the Reduced Triangle Reduce the difficulty of the problem by multiplying or dividing the three lengths by the same number to obtain a similar, but simpler, triangle in the same family 2. Solve for the missing side of this easier triangle. 3. Convert back to the original problem. Example 2 5 55 Notice that 55 and 77 are multiples of 11 We can use 5, x and 7 to find x. 5 and 7 will have much smaller squares than 55 and 77 x 77 7 Don’t forget to multiply x by 11 52 + x2 = 72 25 + x2 = 49 x2 = 24

Example 3 10 24 t t = 26 Example 4 4 3 a Example 5 6 v

Summary Give an example of a Triangle using any 2 whole numbers and explain how to solve that triangle using the Principle of the Reduced Triangle. Homework: Worksheet 9.6