NAME
DATE
8-2
PERIOD
Study Guide and Intervention
The Pythagorean Theorem and Its Converse
The Pythagorean Theorem
In a right triangle, the sum of the
B
c
squares of the lengths of the legs equals the square of the length of
a
the hypotenuse. If the three whole numbers a, b, and c satisfy the equation
A
C
b
a2 + b2 = c2, then the numbers a, b, and c form a
ABC is a right triangle.
Pythagorean triple.
so a2 + b2 = c2.
Example
a. Find a.
b. Find c.
B
a
13
C
12
a2 + b2
a2 + 122
a2 + 144
a2
a
A
=
=
=
=
=
c2
132
169
25
5
20
c
C
30
a2 + b2
202 + 302
400 + 900
1300
√
1300
Pythagorean Theorem
b = 12, c = 13
Simplify.
Subtract.
Take the positive square root
A
=
=
=
=
=
c2
c2
c2
c2
c
Pythagorean Theorem
a = 20, b = 30
Simplify.
Add.
Take the positive square root
of each side.
of each side.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
36.1 ≈ c
Use a calculator.
Exercises
Find x.
1.
3
x
2.
3
9
x
12
5.
4
9
x
25
15
65
x
√
18 or 3 √
2 ≈ 4.2
4.
3.
16
5
9
60
6.
x
x
1
−
√
1345 ≈ 36.7
3
28
11
33
√
663 ≈ 25.7
Use a Pythagorean Triple to find x.
7.
45
8.
9.
8
28
x
x
24
x
96
17
15
Chapter 8
51
100
11
Glencoe Geometry
Lesson 8-2
B
NAME
8-2
DATE
Study Guide and Intervention
PERIOD
(continued)
The Pythagorean Theorem and Its Converse
C
Converse of the Pythagorean Theorem If the sum of the squares
of the lengths of the two shorter sides of a triangle equals the square of
a
b
the lengths of the longest side, then the triangle is a right triangle.
A
B
c
You can also use the lengths of sides to classify a triangle.
If a2 + b2 = c2, then
if a2 + b2 = c2 then ABC is a right triangle.
if a2 + b2 > c2 then ABC is acute.
if a2 + b2 < c2 then ABC is obtuse.
Example
ABC is a right triangle.
Determine whether PQR is a right triangle.
a2 + b2 c2
)2 202
102 + (10 √3
100 + 300 400
400 = 400
Compare c2 and a2 + b2
20
3
a = 10, b = 10 √
3 , c = 20
10√ 3
1
10
2
Simplify.
Add.
Since c2 = and a2 + b2, the triangle is a right triangle.
Exercises
1. 30, 40, 50
yes, right;
502 = 302 + 402
4. 6, 8, 9
yes, acute;
92 < 62 + 82
5 , √
12 , √
13
7. √
yes, acute;
2
2
2
) < ( √5
( √13
) + ( √
12 )
Chapter 8
2. 20, 30, 40
yes, obtuse;
402 > 202 + 302
5. 6, 12, 18
3. 18, 24, 30
yes, right;
302 = 242 + 182
6. 10, 15, 20
no; 6 + 12 = 18
, √
8. 2, √8
12
yes, right;
2
2
( √
12 ) = ( √
8 ) + 22
12
yes, obtuse;
202 > 102 + 152
9. 9, 40, 41
yes, right;
412 = 402 + 92
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine whether each set of measures can be the measures of the sides of a
triangle. If so, classify the triangle as acute, obtuse, or right. Justify your answer.