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?
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.
Definition
Any three whole numbers that satisfy the equation
a2 + b 2 = c 2
form a Pythagorean Triple
Let’s look at some examples
3
8
6
4
10
5
5 3
15
9
12
3 3
4 3
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
(2  3, 2  4, 2  5)
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
2x x  2 x
2  4  84
our work. So lets multiply
value by 2 to get rid of the
11/2
7
fractions
2  7  15
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
1. 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
77
7
Don’t forget to
multiply x by 11
x  11 2 6
x  22 6
Notice that 55 and 77 are multiples of 11
x
We can use 5, x and 7 to find x. 5 and 7
will have much smaller squares than 55
and 77
52 + x2 = 72
25 + x2 = 49
x2 = 24
x2 6
Example 3
10
24
t
t = 26
Example 4
4
3
a 7
a
Example 5
18 3
v
6
v  12 7
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