Lesson 9.6 Families of Right Triangles

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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

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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.
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Definition Any three whole numbers that satisfy
the equation a2 b2 c2
form a Pythagorean Triple
Lets look at some examples
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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.
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The Principle of the Reduced Triangle
Lets 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 .
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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
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
Dont forget to multiply x by 11
52 x2 72
25 x2 49
x2 24
7
Example 3
10
24
t
t 26
Example 4
4
3
a
Example 5
6
v
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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