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Lesson 8-1
The Pythagorean Theorem and Its Converse
Lesson Objectives
1 Use the Pythagorean Theorem
2 Use the Converse of the Pythagorean
Theorem
NAEP 2005 Strand: Geometry
Topic: Relationships Among Geometric Figures
Local Standards: ____________________________________
Vocabulary and Key Concepts.
In a right triangle, the sum of the squares of the lengths of the
is equal to the square of the length of the
a2 b2 c
a
.
b
2
Theorem 8-2: Converse of the Pythagorean Theorem
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Theorem 8-1: Pythagorean Theorem
If the square of the length of one side of a triangle is equal to the sum of
the squares of the lengths of the other two sides, then the triangle is a
triangle.
If the square of the length of the longest side of a triangle is greater than
B
the sum of the squares of the lengths of the other two sides, the triangle
is
c
a
.
C
If c 2 a 2 b 2, the triangle is
A
b
.
Theorem 8-4
B
If the square of the length of the longest side of a triangle is less than
the sum of the squares of the lengths of the other two sides, the triangle
is
a
c
C
b
.
If c 2 a 2 b 2, the triangle is
A
.
A Pythagorean triple is
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Geometry Lesson 8-1
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Theorem 8-3
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Example.
1 Pythagorean Triples Find the length of the hypotenuse of #ABC.
B
Do the lengths of the sides of #ABC form a Pythagorean triple?
a2 1 b2 5 c2
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Î
Use the
Theorem.
c2
Substitute
c2
Simplify.
c
Take the square root.
c
Simplify.
The length of the hypotenuse is
form a
for a, and
12
for b.
A
. The lengths of the sides, 5, 12, and
because they are
5
C
,
numbers that
satisfy a2 1 b2 5 c2.
Quick Check.
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
1. A right triangle has a hypotenuse of length 25 and a leg of length 10. Find the
length of the other leg. Do the lengths of the sides form a Pythagorean triple?
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Geometry Lesson 8-1
143
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Example.
2 Using Simplest Radical Form Find the value of b. Leave your
answer in simplest radical form
1
b2
5
c2
Use the
b2 Substitute
Simplify.
b2
b2 b Î
b Î
12
Subract
8
Theorem.
for a,
for c.
b
from both sides.
Take the square root.
(
)
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a2
Simplify.
b
Quick Check.
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
2. The hypotenuse of a right triangle has length 12. One leg has length 6. Find the
length of the other leg. Leave your answer in simplest radical form.
144
Geometry Lesson 8-1
Daily Notetaking Guide
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Examples.
3 Is It a Right Triangle? Is this triangle a right triangle?
4m
a2 1 b2 0 c2
0
Substitute
0
Simplify.
for a,
for b, and
7m
for c.
6m
2
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Because a2 b2
c2, the triangle
a right triangle.
4 Classifying Triangles as Acute, Obtuse, or Right The numbers 10, 15,
and 20 represent the lengths of the sides of a triangle. Classify the triangle
as acute, obtuse, or right.
c2 0 a2 1 b2
Compare c 2 with a 2 b 2.
202 0
Substitute the greatest length for
0
Simplify.
.
.
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
Because c2
a2 b2, the triangle is a(n)
triangle.
Quick Check.
3. A triangle has sides of lengths 16, 48, and 50. Is the triangle a right triangle?
4. A triangle has sides of lengths 7, 8, and 9. Classify the triangle by its angles.
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Geometry Lesson 8-1
145