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The Pythagorean Theorem
BEFORE
Now
WHY?
You classified triangles.
You’ll find the length of a
side of a right triangle.
So you can find the lengths of
exit ramps, as in Exs. 16–17.
In the Real World
Televisions A television screen is
rectangular and is measured by the
length of a diagonal. The screen at
the right is 15 inches by 20 inches.
What is the length of the diagonal?
Word Watch
hypotenuse, p. 546
leg, p. 546
Pythagorean theorem,
p. 546
In a right triangle, the side opposite
the right angle is the hypotenuse .
The hypotenuse is the longest side
of a right triangle. The two sides that
form the right angle in a right triangle
are legs . The lengths of the legs and
the hypotenuse of any right triangle
are related by the Pythagorean theorem.
15 in.
20 in.
hypotenuse
leg
leg
Pythagorean Theorem
Words For any right triangle, the sum
of the squares of the lengths of the
legs equals the square of the length
of the hypotenuse.
c
a
b
Algebra a 2 b 2 c 2
EXAMPLE
1
Finding the Length of a Hypotenuse
To find c, the length of the television’s diagonal, as described above,
use the Pythagorean theorem. Let a 15 and b 20.
with
a2 b2 c2
Solving
Take the positive square
root in Example 1 because
length is never negative.
152 202 c 2
625 c 2
625
c
25 c
Write Pythagorean theorem.
Substitute 15 for a and 20 for b.
Simplify.
Take positive square root of each side.
Evaluate square root.
ANSWER The length of the television’s diagonal is 25 inches.
546
Chapter 11
Measurement and Area
Page 2 of 4
2
EXAMPLE
Approximating the Length of a Hypotenuse
For the right triangle shown, find the length
of the hypotenuse to the nearest tenth.
with
Review
Need help with
approximating square
roots? See p. 540.
a2 b2 c2
Write Pythagorean theorem.
82 82 c 2
Substitute 8 for a and 8 for b.
8 mm
2
128 c
c
8 mm
Simplify.
128
c
Take positive square root of each side.
11.3 ≈ c
Approximate square root.
ANSWER The length of the hypotenuse is about 11.3 millimeters.
Your turn now
Find the length of the hypotenuse. Round to the
nearest tenth if necessary.
1.
2.
c
3.
12 m
11 cm
5m
9 ft
c
c
9 cm
12 ft
3
EXAMPLE
Finding the Length of a Leg
Camping You are setting up a tent. The ropes
that are used to hold the tent down are 5 feet
long, and the roof of the tent ends 3 feet
above the ground. What is the farthest 5 ft
distance from the base of the tent
that the rope can be staked down?
b
Solution
a2 b 2 c 2
2
2
2
3 b 5
2
9 b 25
b 2 16
Write Pythagorean theorem.
Substitute 3 for a and 5 for c.
Evaluate powers.
Subtract 9 from each side.
b 16
Take positive square root of each side.
b4
Evaluate square root.
ANSWER The farthest distance that the rope can be staked down is
4 feet from the base of the tent.
Lesson 11.3
The Pythagorean Theorem
547
Page 3 of 4
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 715
Getting Ready to Practice
1. Vocabulary Copy and complete: The _?_ is the side opposite the right
angle in a right triangle.
Find the unknown length. Round to the nearest tenth if necessary.
2.
b
3.
c
4.
7 yd
5 cm
a
12 ft
1 yd
20 ft
10 cm
5. Kite Find the wingspan of the kite
height
shown if the height is perpendicular
to the wingspan. Round to the
nearest tenth.
1m
0.7 m
wingspan
Practice and Problem Solving
with
Example
1
2
3
Homework
Exercises
6–12
6–17
6–12
Find the unknown length. Round to the nearest tenth if necessary.
6.
7.
5 in.
c
8.
1.2 cm
13 yd
a
7 in.
0.9 cm
c
9 yd
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
A right triangle has legs of length a and b and hypotenuse of length c.
Find the unknown length. Round to the nearest tenth if necessary.
9. b 1.5 mm, c 2.5 mm
11. a 10 ft, b 24 ft
10. b 11 in., c 13 in.
12. a 7 cm, c 11.2 cm
13. Compare Measure the length and width of a rectangular room. Use the
Pythagorean theorem to calculate the length of a diagonal. Then
measure the length of the diagonal and compare with your calculation.
Tiling In Exercises 14 and 15, use the following
information. You are tiling a floor. You cut several
1 foot by 1 foot tiles along a diagonal as shown.
diagonal
1 ft
14. Find the length of a diagonal of a tile to the
nearest tenth of a foot.
1 ft
15. How many whole diagonal edges will fit along a wall that is 6 feet in
length? along a wall that is 8 feet in length?
548
Chapter 11
Measurement and Area
Page 4 of 4
Handicap Exit Ramps In Exercises 16 and 17, use the following
information. According to federal construction codes, exit ramps
should decrease 1 foot in height every 12 feet in horizontal length.
Height
Length of ramp
16. Find the length of the ramp to the nearest tenth if the ramp has
a height of 1 foot.
17. Find the length of the ramp to the nearest tenth if the ramp has
a height of 1.5 feet.
18. Describe What happens to the length of the hypotenuse when you
double the lengths of the legs of a right triangle? Give an example to
support your conclusion.
19. Challenge The length of the hypotenuse of an isosceles right triangle
is 32 m. Find the length of the legs. Round to the nearest tenth.
Mixed Review
Find the measure of an angle complementary to
20. m
A 18
A 59.2
21. m
A. (Lesson 10.1)
22. m
A 78.4
Order the numbers from least to greatest. (Lesson 11.2)
23. 81
, 72
, 9.2, 8.6
1
24. 1
, , 2.5, 2
3
2
25. 1 , 3
, 0.3, 1.3
5
Basic Skills Find the perimeter and the area of the rectangle
with the given dimensions.
26. l 14 in., w 8 in. 27. l 25 m, w 10 m 28. l 12 ft, w 15 ft
Test-Taking Practice
INTERNET
State Test Practice
CLASSZONE.COM
29. Multiple Choice The length of a leg of a right triangle is 12 meters, and
the hypotenuse is 20 meters. What is the length of the other leg?
A. 8 m
B. 12 m
C. 16 m
D. 23.3 m
30. Multiple Choice An airplane leaves a
runway and travels 1105 feet through
the air. It travels a ground distance of
1100 feet. What is the altitude of the
airplane at this moment?
F. 5 ft
H. 105 ft
1105 ft
Altitude
1100 ft
G. 25 ft
I. 1559 ft
Lesson 11.3
Not drawn to scale
The Pythagorean Theorem
549