Page 1 of 7
10.3
Goal
Find the side lengths of
30-60-90 triangles.
Key Words
• 30-60-90 triangle
30-60-90 Triangles
A right triangle with angle measures of 30, 60, and 90
is called a 30-60-90 triangle .
2 30
b
Activity 10.3 shows that the ratio of the length
of the hypotenuse of a 30-60-90 triangle
to the length of the shorter leg is 2 : 1.
P
60
1
R
Find Leg Length
1
EXAMPLE
P
In the diagram above, T PQR is a 30-60-90 triangle
with PQ 2 and PR 1. Find the value of b.
Solution
You can use the Pythagorean Theorem to find the value of b.
(leg)2 (leg)2 (hypotenuse)2
2
2
Write the Pythagorean Theorem.
2
1 b 2
Substitute.
2
1b 4
Simplify.
b2 3
Subtract 1 from each side.
b 3
Visualize It!
Take the square root of each side.
Because all 30-60-90 triangles are similar, the ratio of the length
of the longer leg to the length of the shorter leg is always 3
: 1.
This result is summarized in the theorem below.
30 30
30
60
60
10
8
An equilateral triangle
can be divided into two
30°-60°-90° triangles.
4 3
60
4
30 53
12 30 63
60
6
60
5
THEOREM 10.2
30-60-90 Triangle Theorem
Words
In a 30-60-90 triangle, the hypotenuse
is twice as long as the shorter leg, and
the longer leg is the length of the shorter
leg times 3
.
Symbols
2x 30 x 3
60
x
Hypotenuse 2 p shorter leg
Longer leg shorter leg p 3
10.3
30-60 -90 Triangles
549
Page 2 of 7
EXAMPLE
2
Find Hypotenuse Length
In the 30-60-90 triangle at the right,
the length of the shorter leg is given.
Find the length of the hypotenuse.
60
12
30
Solution
The hypotenuse of a 30-60-90 triangle is twice as long as
the shorter leg.
hypotenuse 2 p shorter leg
ANSWER
30-60-90 Triangle Theorem
2 p 12
Substitute.
24
Simplify.
The length of the hypotenuse is 24.
EXAMPLE
3
Find Longer Leg Length
In the 30-60-90 triangle at the right,
the length of the shorter leg is given.
Find the length of the longer leg.
5
60
30
Solution
The length of the longer leg of a 30-60-90 triangle is the length of
the shorter leg times 3
.
longer leg shorter leg p 3
5 p 3
ANSWER
30-60-90 Triangle Theorem
Substitute.
The length of the longer leg is 53
.
In a 30-60-90 triangle, the longer leg is opposite the 60 angle, and
the shorter leg is opposite the 30 angle.
Find Lengths in a Triangle
Find the value of x. Write your answer in radical form.
1.
2.
x 30
3.
3
30
x
60
7
550
Chapter 10
Right Triangles and Trigonometry
10
60
60
x
30
Page 3 of 7
EXAMPLE
4
Find Shorter Leg Length
In the 30-60-90 triangle at the right,
the length of the longer leg is given.
Find the length x of the shorter leg.
Round your answer to the nearest tenth.
60 x
30
5
Solution
The length of the longer leg of a 30-60-90 triangle
.
is the length of the shorter leg times 3
longer leg shorter leg p 3
5 x p 3
ANSWER
IStudent Help
30-60-90 Triangle Theorem
Substitute.
5
x
3
Divide each side by 3
.
2.9 ≈ x
Use a calculator.
The length of the shorter leg is about 2.9.
EXAMPLE
5
Find Leg Lengths
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
In the 30-60-90 triangle at the right, the length
of the hypotenuse is given. Find the length x of
the shorter leg and the length y of the longer leg.
y
30
8
60
x
Solution
Use the 30-60-90 Triangle Theorem to find the length of the
shorter leg. Then use that value to find the length of the longer leg.
Shorter leg
Longer leg
hypotenuse 2 p shorter leg
longer leg shorter leg p 3
82px
y 4 p 3
4x
y 43
ANSWER
The length of the shorter leg is 4.
.
The length of the longer leg is 43
Find Leg Lengths
Find the value of each variable. Round your answer to the nearest tenth.
4.
5.
60 x
y 30 42
30
6
60
x
10.3
30-60 -90 Triangles
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Page 4 of 7
10.3 Exercises
Guided Practice
Vocabulary Check
1. Name two special right triangles by their angle measures.
Skill Check
Use the diagram to tell whether the equation is true or false.
2. t 73
3. t 3
h
4. h 2t
5. h 14
h
6. 7 2
t
7. 7 3
60
h
30
t
Find the value of each variable. Write your answers in radical form.
8.
9.
60
x
h
5
10.
30
30
y
a
8
60
60
4
30
b
Practice and Applications
Extra Practice
Finding Hypotenuse Lengths Find the length of the hypotenuse.
See p. 693.
11.
12.
3
60
13.
60
x
x
14.
60
x
60
15.
x
8
30
30
113
30
16.
9
x 30
30
30 x
123
60
60
1
Finding Leg Lengths Find the length of the longer leg of the
triangle. Write your answer in radical form.
17.
18.
20.
Example 2:
Example 3:
Example 4:
Example 5:
552
13
x
x
21.
60
22.
30
Exs. 11–16
Exs. 17–22
Exs. 23–28
Exs. 29–35
Chapter 10
30
6
30
30
x
Homework Help
x
19.
60
4 60
x 30
60
20
Right Triangles and Trigonometry
16
30 x
60
60
23
7
Page 5 of 7
Finding Leg Lengths Find the length of the shorter leg of the
triangle. Round your answer to the nearest tenth.
23.
7
24.
60 x
25.
x 60
30
x
30
30
18
60
4
26.
27.
28.
143
30
30 10
60
12 30
x
60
x
60
x
29. Tipping Platform A tipping platform is a ramp used to unload
trucks as shown below. What is the height of an 60 foot ramp
when it is tipped up to a 30 angle?
ramp
height
of ramp
30
60 ft
Finding Leg Lengths Find the value of each variable. Write your
answers in radical form.
30.
Careers
y
30
10
33.
31.
60
32.
x
x
60
8
60
30
30
y
y
34.
y 30 9
60
x
12
x
x
60
35.
80 30 y
y
30 15
60
x
36. Fitness A “crunch” is the type
PERSONAL TRAINERS
develop fitness programs
suited to an individual’s
abilities and goals. They
study anatomy, nutrition,
and physiology.
of sit-up shown in the photo at
the right. A personal trainer tells
you that in doing a crunch, your
back and shoulders should be
lifted to an angle of about 30.
If your shoulder-to-waist length
is 18 inches, how high should
your shoulders be lifted?
Career Links
CLASSZONE.COM
10.3
30-60 -90 Triangles
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Page 6 of 7
Find Area Using 30-60-90 Triangles
EXAMPLE
The road sign is shaped like an equilateral
triangle with side lengths of 36 inches.
Estimate the area of the sign.
Solution
18 in.
Divide the triangle into two 30-60-90 triangles.
Student Help
The length of the shorter leg of each triangle is
18 inches. The length of the longer leg of each triangle
is 183
inches, by the 30-60-90 Triangle Theorem.
LOOK BACK
Use the formula for the area of a triangle.
To review the area of a
triangle, see p. 431.
Area bh p 36 p 183
≈ 561.18
1
2
ANSWER
h
1
2
The area of the sign is about 561 square inches.
Finding Area Find the area of each triangle. Use the example above
as a model.
37.
38.
7 cm
39.
18 in.
30 ft
Using Algebra Use the figure in the example above to explain
40.
why the area A of an equilateral triangle with side length s is given
1
4
by the formula A p s 2 p 3
.
41.
Visualize It! A 30-60-90 triangle has a shorter leg length of
15 centimeters. Sketch the triangle and find the length of the
hypotenuse and the length of the longer leg in radical form.
42. Challenge The side length of the
1 cm
hexagonal nut shown at the right
is 1 centimeter. Find the value of x.
(Hint: Use the fact that a regular
hexagon can be divided into six
congruent equilateral triangles.)
Standardized Test
Practice
43. Multiple Choice Which triangle is labeled correctly?
A
B
8
10
6
554
Chapter 10
x
Right Triangles and Trigonometry
C
143
60
7
14
10
30
53
5
D
8
16
60
83
36 in.
Page 7 of 7
44. Multiple Choice Find the perimeter of the triangle shown below
to the nearest tenth of a centimeter.
F
H
Mixed Review
G
J
28.4 cm
31.2 cm
30 cm
12 cm
41.6 cm
30
Writing Ratios A football team won 10 games and lost 6 games.
Find the ratio. (Lesson 7.1)
45. wins to losses
46. losses to wins
47. wins to the number of games
48. losses to the number of games
Similar Triangles Determine whether the triangles are similar. If they
are similar, write a similarity statement. (Lessons 7.3, 7.4)
A
49.
50.
58
8
37
B
S
C F
85
H
Algebra Skills
P
P
14
R
12
58
G
21
T
U
Evaluating Expressions Evaluate the expression when x 4.
(Skills Review, p. 670)
51. 5x 4
52. 10x 1
53. x 2 7
54. (x 3)(x 3)
55. 2x 2 x 1
56. 5x 2 2x 3
Quiz 1
Multiply the radical expression. Then simplify if possible.
(Lesson 10.1)
1. 8
p 3
2. 2
p 1
5
3. 8
p 1
8
4. 8
0
p 5
Simplify the radical expression. (Lesson 10.1)
5. 2
7
6. 1
7
6
7. 5
2
8. 1
8
0
Find the value of each variable. Write your answer in radical form.
(Lessons 10.2, 10.3)
9.
10.
2 30
11.
3
3
y
30
28
x
60
x
60
x
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30-60 -90 Triangles
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