Page 1 of 7
4.4
Goal
Use the Pythagorean
Theorem and the Distance
Formula.
The Pythagorean Theorem
and the Distance Formula
The photo shows part of twin skyscrapers
in Malaysia that are connected by a
skywalk. The skywalk is supported by a
set of beams.
leg
In a right triangle, the sides that form
the right angle are called legs .
The side opposite the right angle is
called the hypotenuse .
Key Words
• leg
• hypotenuse
• Pythagorean Theorem
• Distance Formula
leg
hypotenuse
In Exercise 26 you will find the length
of the support beam using the
Pythagorean Theorem.
THEOREM 4.7
The Pythagorean Theorem
Words
In a right triangle, the square of the
length of the hypotenuse is equal to
the sum of the squares of the lengths
of the legs.
EXAMPLE
c
a
C
(hypotenuse)2 (leg)2 (leg)2
Symbols
B
b
If maC 90, then c 2 a2 b2.
Find the Length of the Hypotenuse
1
Find the length of the hypotenuse.
12
5
c
Solution
(hypotenuse)2 (leg)2 (leg)2
2
2
c 5 12
2
c 2 25 144
Student Help
2
c 169
c2 169
SKILLS REVIEW
To review square roots,
see p. 669.
c 13
ANSWER
192
Chapter 4
Triangle Relationships
Pythagorean Theorem
Substitute.
Multiply.
Add.
Find the positive square root.
Solve for c.
The length of the hypotenuse is 13.
A
Page 2 of 7
Find the Length of a Leg
2
EXAMPLE
Find the unknown side length.
b
7
14
Solution
(hypotenuse)2 (leg)2 (leg)2
2
2
14 7 b
Pythagorean Theorem
2
Substitute.
2
Multiply.
2
Subtract 49 from each side.
196 49 b
196 49 49 b 49
147 b 2
Simplify.
1
4
7
b
2
Find the positive square root.
12.1 ≈ b
ANSWER
Approximate with a calculator.
The side length is about 12.1.
Find the Lengths of the Hypotenuse and Legs
Find the unknown side length.
1.
2.
3.
15
b
17
a
EXAMPLE
c
Find the Length of a Segment
3
Find the distance between the points
A(1, 2) and B(4, 6).
Student Help
LOOK BACK
To review finding
distances on a
coordinate plane,
see p. 30.
8
7
10
6
y
B (4, 6)
A(1, 2)
C(4, 2)
Solution
&*.
Draw a right triangle with hypotenuse AB
BC 6 2 4 and CA 4 1 3. Use the
Pythagorean Theorem.
1
1
x
(hypotenuse)2 (leg)2 (leg)2
(AB)2 32 42
Substitute.
2
Multiply.
2
Add.
(AB) 9 16
(AB) 25
(A
B
)2 2
5
AB 5
4.4
Find the positive square root.
Simplify.
The Pythagorean Theorem and the Distance Formula
193
Page 3 of 7
Distance Formula Using the steps shown in Example 3, the
Pythagorean Theorem can be used to develop the Distance Formula ,
which gives the distance between two points in a coordinate plane.
THE DISTANCE FORMULA
If A(x1, y1) and B(x2, y2) are
points in a coordinate plane,
then the distance between
A and B is
y
AB (x
x1)2(y
y
)2.
2
2
1
A(x1, y1)
B(x2, y2)
y2 y1
C (x2, y1)
x2 x1
x
4
EXAMPLE
Use the Distance Formula
Find the distance between D(1, 2) and E(3, 2).
y D(1, 2)
1
Solution
1
Begin by plotting the points in a coordinate plane.
x
x1 1, y1 2, x2 3, and y2 2.
E(3, 2)
Student Help
STUDY TIP
DE (x
x
)2(y2
y
)2
2
1
1
46
1 4 16
.
The square root of a
sum does not equal
the sum of the square
roots. You must add 4
and 16 before taking
the square root.
The Distance Formula
(3
1
)2
(
2
2
)2
Substitute.
2
(
4
)
Simplify.
4
6
1
Multiply.
2
0
Add.
≈ 4.5
Approximate with a calculator.
2
ANSWER
2
The distance between D and E is about 4.5 units.
Use the Distance Formula
Find the distance between the points.
4.
5.
y
6.
y D(1, 4)
y
F (2, 2)
B (3, 4)
1
1
1
A(0, 0)
194
Chapter 4
1
1
Triangle Relationships
4
x
x
E(3, 2)
G(3, 3)
x
Page 4 of 7
4.4 Exercises
Guided Practice
Vocabulary Check
1. Sketch a right triangle and label its vertices. Then use your
triangle to state the Pythagorean Theorem.
Skill Check
Find the unknown side length. Round your answer to the nearest
tenth, if necessary.
x
2.
3.
4.
x
x
4
1
8
10
8
2
Find the distance between the points. Round your answer to the
nearest tenth, if necessary.
5.
6.
y
y
7.
D(4, 6)
y
B(5, 3)
G(3, 3)
1
1
1
4
A(0, 0)
1
x
x
C (2, 1)
1
F(1, 3)
x
Practice and Applications
Extra Practice
See p. 681.
Finding a Hypotenuse Find the length of the hypotenuse.
8.
9.
10.
c
c
9
c
65
40
9
72
12
11.
12.
24
10
13.
c
8
35
c
15
12
c
Homework Help
Example 1: Exs. 8–13,
26
Example 2: Exs. 14–22
Example 3: Exs. 27–34
Example 4: Exs. 27–34
Finding a Leg Find the unknown side length.
14.
25
b
24
15.
a
89
16.
39
5
61
b
4.4
The Pythagorean Theorem and the Distance Formula
195
Page 5 of 7
Pythagorean Triples
EXAMPLE
A Pythagorean triple is a set of three positive integers a, b, and c that
satisfy the equation c 2 a 2 b 2. For example, the integers 3, 4, and 5
form a Pythagorean triple because 52 32 42.
Find the length of the hypotenuse of the right
triangle. Tell whether the side lengths form a
Pythagorean triple.
c
8
15
Solution
(hypotenuse) 2 (leg)2 (leg)2
2
2
c 8 15
Pythagorean Theorem
2
Substitute 8 and 15 for the legs.
2
c 64 225
Multiply.
c 2 289
Add.
c 17
ANSWER
Find the positive square root.
Because the side lengths 8, 15, and 17 are integers,
they form a Pythagorean triple.
Pythagorean Triples Find the unknown side length. Tell whether the
side lengths form a Pythagorean triple.
17.
18.
b
19.
6
c
7
3
Architecture
4
5
c
11
20.
30
16
21.
24
22.
50
9
48
b
c
a
Visualize It! Tell whether the side lengths form a Pythagorean triple.
If so, draw a right triangle with the side lengths.
23. 21, 29, 20
PETRONAS TOWERS
These 1483 foot buildings
tower over the city of Kuala
Lumpur, Malaysia. When the
Petronas Towers were
designed by Cesar Pelli in
1991, they were the tallest
buildings in the world.
shown on page 170 are connected
by a skywalk with support beams.
Use the diagram to find the
approximate length of each
support beam.
Application Links
Chapter 4
25. 5, 12, 14
26. Support Beam The skyscrapers
CLASSZONE.COM
196
24. 25, 7, 24
Triangle Relationships
x
x
support
beams
Page 6 of 7
Distance Formula Find the distance between the points. Round your
answer to the nearest tenth.
27.
28.
y
29.
y
y
F(5, 5)
B(5, 2)
D(6, 8)
1
1
x
1
2
A(2, 2)
E(4, 1)
C (0, 2)
2
ICLASSZONE.COM
HOMEWORK HELP
Extra help with problem
solving in Exs. 30–32 is
at classzone.com
Congruence Graph P, Q, and R. Then use the Distance Formula to
&* c QR
&*.
decide whether PQ
30. P(4, 4)
31. P(1, 6)
Q(1, 6)
R(1, 3)
32. P(5, 1)
Q(5, 7)
R(3, 6)
Q(8, 5)
R(3, 2)
Sum of Distances In Exercises 33 and 34, use the map below.
Sidewalks around the edge of a campus quadrangle connect the
buildings. Students sometimes take shortcuts by walking across the
grass along the pathways shown. The coordinate system shown is
measured in yards.
dorm
dorm
F (0, 30)
B (65, 30)
E (100, 30)
dorm
library
IStudent Help
5 x
x
C (0, 15)
A(50, 0)
G (0, 0)
D (100, 0)
dorm
classroom
dorm
33. Find the distances from A to B, from B to C, and from C to A if you
have to walk around the quadrangle along the sidewalks.
34. Find the distances from A to B, from B to C, and from C to A if you
are able to walk across the grass along the pathways.
Challenge Find the value of x. Use a calculator, and round your
answer to the nearest tenth.
35.
36.
37.
x
x
10
6
5
x
3
11
8
12
8
4.4
The Pythagorean Theorem and the Distance Formula
197
Page 7 of 7
Standardized Test
Practice
38. Multiple Choice What is the distance from (3, 5) to (1, 4)?
A
B
5
C
1
7
D
21
3
9
7
39. Multiple Choice Which of the following is the length of the
hypotenuse of a right triangle with legs of lengths 33 and 56?
F
Mixed Review
G
65
H
72.9
J
85.8
89
Finding Absolute Values Evaluate. (Skills Review, p. 662)
40.
7
41.
1.05
42.
0
43.
0.02
Solving Inequalities Solve the inequality. (Algebra Review, p. 167)
44. x 5 < 8
Algebra Skills
45. 10 x ≥ 12
46. 4x ≥ 28
47. 6x 11 ≤ 11
Fractions and Decimals Write the decimal as a fraction in simplest
form. (Skills Review, p. 657)
48. 0.4
49. 0.08
50. 0.54
51. 0.12
52. 0.250
53. 0.173
54. 0.3
xx
55. 0.1
xx
Quiz 2
Find the value of x. (Lesson 4.3)
1.
2.
3x 5
13
3.
x5
55
(2x 1)
4x 16
In Exercises 4–6, find the distance between the points. (Lesson 4.4)
4.
5.
y
6.
y
y
1
A(2, 3)
A(3, 1)
1
B(1, 2)
1
x
B(1, 2)
1 B (3, 0) x
1
1
A(1, 1)
7. A device used to measure windspeed is
attached to the top of a pole. Support wires
are attached to the pole 5 feet above the
ground. Each support wire is 6 feet long.
About how far from the base of the pole
is each wire attached to the ground?
(Lesson 4.4)
198
Chapter 4
Triangle Relationships
6 ft
5 ft
x