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Transparency 7-2
5-Minute Check on Lesson 7-1
Find the geometric mean between each pair of numbers. State exact
answers and answers to the nearest tenth.
1. 9 and 13
2. 2√5 and 5√5
√117 ≈ 10.8
3. Find the altitude
4. Find x, y, and z
y
8
3
x
z
x = 8,
y = √80 ≈ 8.9
z = √320 ≈ 17.9
4
√24 ≈ 4.9
5.
√50 ≈ 7.1
20
13
Which of the
following is the best estimate of x?
Standardized Test Practice:
x
5
A
2
B
10
C
11
D
12
Click the mouse button or press the
Space Bar to display the answers.
12
Lesson 7-2
Pythagorean Theorem and its
Converse
Objectives
• Use the Pythagorean Theorem
– If a right triangle, then c² = a² + b²
• Use the converse of the Pythagorean Theorem
– If c² = a² + b², then a right triangle
Vocabulary
• None new
Pythagorean Theorem
a
c
Pythagorean Theorem
a2 + b2 = c2
Sum of the squares of the legs is equal
to the square of the hypotenuse
b
Converse of the Pythagorean Theorem: If the sum of the squares
of the measures of two sides of a triangle equals the square of the
measure of the longest side, then the triangle is a right triangle
Remember from our first computer quiz:
In an acute triangle, c2 < a2 + b2.
In an obtuse triangle, c2 > a2 + b2.
Example 1
Find d.
Pythagorean Theorem
Simplify.
Subtract 9 from each side.
Take the square root of each side.
Use a calculator.
Answer:
Example 2
Find x.
Answer:
Pythagorean Triples
For three numbers to be a Pythagorean triple they
must satisfy both of the following conditions:
–
–
They must satisfy c2 = a2 + b2
where c is the largest number
All three must be whole numbers (integers)
Common Pythagorean Triples:
3, 4, 5
5, 12, 13
8, 15, 17
6, 8, 10
9, 12, 15
7, 24, 25
12, 16, 20
9, 40, 41
15, 20, 25
10, 24, 26
16, 30, 34
Example 3
Determine whether 9, 12, and 15 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Since the measure of the longest side is 15, 15 must be c.
Let a and b be 9 and 12.
Pythagorean Theorem
Simplify.
Add.
Answer: These segments form the sides of a right triangle
since they satisfy the Pythagorean Theorem. The measures
are whole numbers and form a Pythagorean triple.
Example 4
Determine whether 21, 42, and 54 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Pythagorean Theorem
Simplify.
Add.
Answer: Since
, segments with these
measures cannot form a right triangle. Therefore,
they do not form a Pythagorean triple.
Example 5
Determine whether
4, and 8 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Pythagorean Theorem
Simplify.
Add.
Answer: Since 64 = 64, segments with these measures
form a right triangle. However,
is not a whole number.
Therefore, they do not form a Pythagorean triple.
Example 6
Determine whether each set of measures are the sides
of a right triangle. Then state whether they form a
Pythagorean triple.
a. 6, 8, 10
Answer: The segments form the sides of a right triangle
and the measures form a Pythagorean triple.
b. 5, 8, 9
Answer: The segments do not form the sides of a right
triangle, and the measures do not form a Pythagorean triple.
c.
Answer: The segments form the sides of a right triangle,
but the measures do not form a Pythagorean triple.
Summary & Homework
• Summary:
– The Pythagorean Theorem can be used to find the
measures of the sides of a right triangle
– If the measures of the sides of a triangle form a
Pythagorean triple, then the triangle is a right
triangle
• Homework:
– pg 354, 17-19, 22-25, 30-35