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Chapter 3 l Skills Practice 351
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Skills Practice Skills Practice for Lesson 3.1
Name _____________________________________________ Date ____________________
Get Radical or (Be)2!Radicals and the Pythagorean Theorem
Vocabulary Write the term that best completes each statement.
1. An expression that includes a symbol such as !__
A or 3 !"" A is called a(n) radical expression .
2. A quantity that is enclosed by a radical symbol is called the radicand .
3. The process of eliminating a radical from the denominator is called rationalizing the denominator .
4. The Pythagorean Theorem states that a2 ! b2 " c2, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse of the triangle.
Problem Set Simplify each radical expression completely.
1. !___
27 2. !___
50
!______
9 ! 3 " !__
9 !__
3 " 3 !__
3 !_______
25 ! 2 " !___
25 !__
2 " 5 !__
2
3. !____
128 4. !____
112
!_______
64 ! 2 " !___
64 !__
2 " 8 !__
2 !_______
16 ! 7 " !___
16 !__
7 " 4 !__
7
5. 4 !___
18 6. 6 !___
72
4 !______
9 ! 2 " 4 !__
9 !__
2 " 12 !__
2 6 !_______
36 ! 2 " 6 !___
36 !__
2 " 36 !__
2
7. !____
a2b5 8. !____
x4y3
!______
a2b4b " !__
a2 !___
b4 !__
b " ab2 !__
b !_____
x4y2y " !__
x4 !__
y2 !__
y " x2y !__
y
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9. !_____
9x7y2
!______
9x6xy2 " !__
9 !__
x6 !__
x !__
y2 " 3x3y !__
x
10. !______
24x3y5
!___________
(4)(6)x2xy4y " !__
4 !__
6 !__
x2 !__
x !__
y4 !__
y " 2 xy2 !____
6xy
Rationalize the denominator to simplify each radical expression completely.
11. 3 ___ !
__ 5
3 ___ !
__ 5 " 3 ___
!__
5 # !
__ 5 ___
!__
5 " 3 !
__ 5 ____ 5
12. 5 ___ !
__ 6
5 ___ !
__ 6 " 5 ___
!__
6 # !
__ 6 ___
!__
6 " 5 !
__ 6 ____ 6
13. 6 ___ !
__ 3
6 ___ !
__ 3 " 6 ___
!__
3 # !
__ 3 ___
!__
3 " 6 !
__ 3 ____
3 " 2 !
__ 3
14. 15 ___ !
__ 5
15 ___ !
__ 5 " 15 ___
!__
5 # !
__ 5 ___
!__
5 " 15 !
__ 5 _____ 5 " 3 !
__ 5
15. 2 !__
3 ____ !
__ 6
2 !__
3 ____ !
__ 6 " 2 !
__ 3 ____
!__
6 # !
__ 6 ___
!__
6 " 2 !
__ 3 !
__ 6 ______ 6 "
2 !__
3 !_____
(3)(2) __________ 6 " 2 !
__ 3 !
__ 3 !
__ 2 _________ 6 " 2(3) !
__ 2 ______ 6 " !
__ 2
16. 2 !__
2 ____ !
__ 8
2 !__
2 ____ !
__ 8 " 2 !
__ 2 ____
!__
8 # !
__ 8 ___
!__
8 " 2 !
__ 2 !
__ 4 !
__ 2 _________ 8 " (2)(2)(2) ________ 8 " 8 __ 8 " 1
17. 2ab ____ b !__
a
2ab ____ b !__
a " 2ab ____ b !__
a # !
__ a ___ !
__ a " 2ab !
__ a _______ ab " 2 !
__ a
18. 3 !___
ab _____ !
___ 5a
3 !___
ab _____ !
___ 5a " 3 !
___ ab _____
!___
5a # !
___ 5a ____
!___
5a " 3 !
__ a !
__ b !
__ 5 !
__ a ___________ 5a " 3a !
__ b !
__ 5 ________ 5a " 3 !
___ 5b _____ 5
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Use the Pythagorean Theorem to answer each question.
19. The interior of a small moving van has a height of 9 feet. A couch that is 8 feet long and 2.5 feet high is tipped on its end to fit in the van. Can the couch be set back on its feet while inside the moving van?
a2 $ b2 " c2
(8)2 $ (2.5)2 " c2
64 $ 6.25 " c2
70.25 " c2
!______
70.25 " c
c ! 8.382
Because 8.382 feet is less than 9 feet, the couch can be set back on its feet while inside the moving van.
20. A firefighter has a 22-foot ladder. If he stands the bottom of the ladder 7 feet from the base of a building, will the ladder be long enough to reach a window 19 feet from the ground?
a2 $ b2 " c2
(19)2 $ (7)2 " c2
361 $ 49 " c2
410 " c2
!____
410 " c
c ! 20.248
Because 20.248 feet is less than 22 feet, the ladder will be long enough to reach the window.
21. A baseball diamond is a square with sides of 90 feet. The first base player stands on first base and throws a ball to third base. To the nearest foot, what distance does the ball travel?
a2 $ b2 " c2 The ball travels about 127 feet.
(90)2 $ (90)2 " c2
8100 $ 8100 " c2
16,200 " c2
!_______
16,200 " c
c ! 127.279
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22. A natural gas line is buried diagonally across a rectangular lot. The lot measures 240 feet by 162 feet. To the nearest foot, how much gas line is buried in the lot?
a2 $ b2 " c2 About 290 feet of gas line is buried in the lot.
(240)2 $ (162)2 " c2
57,600 $ 26,244 " c2
83,844 " c2
!_______
83,844 " c
c ! 289.558
23. A guy wire is connected to the top of a 16-meter pole and to a point on the ground 8 meters from the bottom of the pole. To the nearest meter, what length is the guy wire?
a2 $ b2 " c2 The guy wire is about 18 meters long.
(16)2 $ (8)2 " c2
256 $ 64 " c2
320 " c2
!____
320 " c
c ! 17.889
24. Peter lives 10 miles directly north of a tower that broadcasts a wireless signal for his computer. If he drives directly east from his home, he can keep the wireless connection for 7 miles. About how many miles is the broadcasting range of the tower?
a2 $ b2 " c2 The broadcasting range of the tower is about 12 miles.
(10)2 $ (7)2 " c2
100 $ 49 " c2
149 " c2
!____
149 " c
c ! 12.207
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Skills Practice Skills Practice for Lesson 3.2
Name _____________________________________________ Date ____________________
The Pythagorean Theorem Disguised as the Distance Formula!The Distance Formula and Midpoint Formula
Vocabulary Write each formula, explain how it is found, and explain why it is used.
1. the distance formula
The distance formula is used to find the distance between two points. It is derived from the Pythagorean Theorem by drawing a right triangle with the two points as endpoints of the hypotenuse. Using the length of each leg, the length of the hypotenuse can then be found. The distance formula states that the distance between the two points (x1, y1) and (x2, y2) is d ! !
____________________ (x2 " x1)
2 # ( y2 " y1)2 .
2. the midpoint formula
The midpoint formula is used to find the center point between two points on a coordinate plane. It is derived by finding the mean of the x-coordinates and the mean of the y-coordinates. The midpoint formula states that the midpoint
between the two points (x1, y1) and (x2, y2) is ( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) .
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Problem Set Plot each pair of points and connect them with a line segment. Draw a right triangle with this line segment as the hypotenuse. Label the length of all three sides of the right triangle to the nearest tenth.
1. (0, 1) and (8, 5) 2. (2, 4) and (6, 8)y
86 102 4
10
4
6
2
8
!2
!4
!6
!8
!10
!2!4!6!8!10x
0
8.9
8
4
y
86 102 4
10
4
6
2
8
!2
!4
!6
!8
!10
!2!4!6!8!10x
0
5.7 4
4
3. (!1, 3) and (5, 1) 4. (!6, 0) and (2, !4)y
86 102 4
10
4
6
2
8
!2
!4
!6
!8
!10
!2!4!6!8!10x
0
6.3
62
y
86 102 4
10
4
6
2
8
!2
!4
!6
!8
!10
!2!4!6!8!10x
0
8.9
8
4
Chapter 3 l Skills Practice 357
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Use the distance formula to calculate the distance between each pair of points. Round decimal answers to the nearest thousandth.
5. (0, 1) and (8, 5) 6. (2, 4) and (6, 8)
d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2 d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2
d ! !_________________
(8 " 0)2 # (5 " 1)2 d ! !_________________
(6 " 2)2 # (8 " 4)2
d ! !________
64 # 16 d ! !________
16 # 16
d ! !___
80 d ! !___
32
d ! 4 !__
5 ! 8.944 d ! 4 !__
2 ! 5.657
7. (!1, 3) and (5, 1) 8. (!6, 0) and (2, !4)
d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2 d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2
d ! !____________________
(5 " ("1))2 # (1 " 3)2 d ! !______________________
(2 " ("6))2 # ("4 " 0)2
d ! !_______
36 # 4 d ! !________
64 # 16
d ! !___
40 d ! !___
80
d ! 2 !___
10 ! 6.325 d ! 4 !__
5 ! 8.944
9. (!6, !6) and (!6, !12) 10. (!3, 7) and (0, 5)
d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2 d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2
d ! !___________________________
("6 " ("6))2 # ("12 " ("6))2 d ! !____________________
(0 " ("3))2 # (5 " 7)2
d ! !___________
(0)2 # ("6)2 d ! !______
9 # 4
d ! !___
36 d ! !___
13 ! 3.606
d ! 6
Use the given information to solve for y.
11. The distance between (6, 0) and (!3, y) is !____
130 .
d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2
!____
130 ! !___________________
("3 " 6)2 # ( y " 0)2
!____
130 ! !_______
81 # y2
130 ! 81 # y2
49 ! y2
y ! $7
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12. The distance between (!5, 0) and (8, y) is !____
269 .
d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2
!____
269 ! !____________________
(8 " ("5))2 # ( y " 0)2
!____
269 ! !_________
169 # y2
269 ! 169 # y2
100 ! y2
y ! $10
13. The distance between (!4, 0) and (6, y) is !____
116 .
d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2
!____
116 ! !____________________
(6 " ("4))2 # ( y " 0)2
!____
116 ! !_________
100 # y2
116 ! 100 # y2
16 ! y2
y ! $4
14. The distance between (8, 0) and (1, y) is !____
113 .
d ! !____________________
(x2 " x1)2 # ( y2 " y1)
2
!____
113 ! !_________________
(1 " 8)2 # ( y " 0)2
!____
113 ! !_______
49 # y2
113 ! 49 # y2
64 ! y2
y ! $8
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Use the midpoint formula to calculate the midpoint between each pair of points.
15. (0, 1) and (8, 5)
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( 0 # 8 ______
2 , 1 # 5 ______
2 ) ! ( 8 __
2 , 6 __
2 ) ! (4, 3)
16. (2, 4) and (6, 8)
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( 2 # 6 ______
2 , 4 # 8 ______
2 ) ! ( 8 __
2 , 12 ___
2 ) ! (4, 6)
17. (!1, 3) and (5, 1)
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( "1 # 5 _______
2 , 3 # 1 ______
2 ) ! ( 4 __
2 , 4 __
2 ) ! (2, 2)
18. (!6, 0) and (2, !4)
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( "6 # 2 _______
2 , 0 # ("4) ________
2 ) ! ( "4 ___
2 , "4 ___
2 ) ! ("2, "2)
19. (!12, !16) and (!4, !9)
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( "12 # ("4) ___________
2 , "16 # ("9) ___________
2 ) ! ( "16 _____
2 , "25 _____
2 ) ! ("8, "12.5)
20. (!9, !8) and (0, !10)
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( "9 # 0 _______
2 , "8 # ("10) ___________
2 ) ! ( "9 ___
2 , "18 _____
2 ) ! ("4.5, "9)
Use the given information to solve for x.
21. The point (3, 0) is the midpoint of a line segment with endpoints (7, 4) and (x, !4).
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( 7 # x ______
2 , 4 # ("4) ________
2 ) ! ( 7 # x ______
2 , 0 __
2 ) ! ( 7 # x ______
2 , 0 ) ! (3, 0)
7 # x ______ 2 ! 3
7 # x ! 6 x ! "1
22. The point (!6, !2) is the midpoint of a line segment with endpoints (1, 1) and (x, !5).
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( 1 # x ______
2 , 1 # ("5) ________
2 ) ! ( 1 # x ______
2 , "4 ___
2 ) ! ( 1 # x ______
2 , "2 ) ! ("6,"2)
1 # x ______ 2 ! "6
1 # x ! "12 x ! "13
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23. The point (5, !7) is the midpoint of a line segment with endpoints (0, 2) and (x, !16).
( x1 # x2 _______ 2 ,
y1 # y2 _______ 2 ) ! ( 0 # x ______
2 , 2 # ("16) __________
2 ) ! ( x __
2 , "14 _____
2 ) ! ( x __
2 , "7 ) ! (5,"7)
x __ 2 ! 5
x ! 10 24. The point (!8, 4) is the midpoint of a line segment with endpoints (!13, !2) and (x, 10).
( x1 # x2 _______ 2
, y1 # y2 _______
2 ) ! ( "13 # x ________
2 , "2 # 10 ________
2 ) ! ( "13 # x ________
2 , 8 __
2 ) ! ( "13 # x ________
2 , 4 ) ! ("8, 4)
"13 # x ________ 2
! "8
"13 # x ! "16 x ! "3
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Skills Practice Skills Practice for Lesson 3.3
Name _____________________________________________ Date ____________________
Drafting EquipmentProperties of 45º!45º!90º Triangles
Vocabulary Explain how to find each of the following.
1. the leg length of a 45º!45º!90º triangle when you know the length of the other leg
A 45º!45º!90º triangle is an isosceles triangle. So, both legs are the same length. When you know the length of one leg of a 45º!45º!90º triangle, then you also know the other.
2. the leg length of a 45º!45º!90º triangle when you know the length of the hypotenuse
The 45º!45º!90º Triangle Theorem says that the length of the hypotenuse in a 45º!45º!90º triangle is !
__ 2 times the length of a leg. When you know the
length of the hypotenuse and want to find the length of a leg, divide the length of the hypotenuse by !
__ 2 .
3. the hypotenuse of a 45º!45º!90º triangle when you know the length of a leg
The 45º!45º!90º Triangle Theorem says that the length of the hypotenuse in a 45º!45º!90º triangle is !
__ 2 times the length of a leg. When you know the
length of a leg and want to find the length of the hypotenuse, multiply the length of the leg by !
__ 2 .
4. the area of a 45º!45º!90º triangle
You can find the area of any triangle by using the formula A " 1 __ 2 bh, where b is the base length and h is the height. In a 45º!45º!90º triangle, the base and height are the legs, which are the same length. So, to find the area, calculate 1 __ 2 l 2 where l is the length of a leg.
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Problem Set Determine the unknown side length of each triangle.
1.
c
3 m
3 m
2.
4!2 cm
c4!2 cm
c " 3( !__
2 ) " 3 !__
2 m c " 4 !__
2 ( !__
2 ) " 4(2) " 8 cm
3.
18 inches a
45°
45°
4.
11 feet a
45°
45°
a !__
2 " 18 a !__
2 " 11
a " 18 ___ !
__ 2 " 18 ___
!__
2 # !
__ 2 ___
!__
2 " 18 !
__ 2 _____ 2 " 9 !
__ 2 inches a " 11 ___
!__
2 " 11 ___
!__
2 # !
__ 2 ___
!__
2 " 11 !
__ 2 _____ 2 feet
Use the 45º!45º!90º Triangle Theorem to calculate the indicated length.
5. What is the leg length of an isosceles right triangle with a hypotenuse of 6 inches?
Let a " leg length and c " hypotenuse length.
a !__
2 " c
a !__
2 " 6
a " 6 ___ !
__ 2 " 6 ___
!__
2 # !
__ 2 ___
!__
2 " 6 !
__ 2 ____ 2 " 3 !
__ 2
The leg length is 3 !__
2 inches.
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6. What is the leg length of an isosceles right triangle with a hypotenuse of 28 centimeters?
Let a " leg length and c " hypotenuse length.
a !__
2 " c
a !__
2 " 28
a " 28 ___ !
__ 2 " 28 ___
!__
2 # !
__ 2 ___
!__
2 " 28 !
__ 2 _____ 2 " 14 !
__ 2
The leg length is 14 !__
2 centimeters.
7. What is the length of the hypotenuse of an isosceles right triangle with a leg length of 7 feet?
Let a " leg length and c " hypotenuse length.
c " a !__
2
c " 7 !__
2
The length of the hypotenuse is 7 !__
2 feet.
8. What is the length of the hypotenuse of an isosceles right triangle with a leg length of 3 meters?
Let a " leg length and c " hypotenuse length.
c " a !__
2
c " 3 !__
2
The length of the hypotenuse is 3 !__
2 meters.
9. The side length of a square is 4.5 feet. What is the length of the diagonal?
Let a " side length and c " diagonal length.
c " a !__
2
c " 4.5 !__
2
The length of the diagonal is 4.5 !__
2 feet.
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10. The side length of a square is 2 !__
2 feet. What is the length of the diagonal?
Let a " side length and c " diagonal length.
c " a !__
2
c " (2 !__
2 )( !__
2 )
c " 2(2) " 4
The length of the diagonal is 4 feet.
11. The length of the diagonal of a square is 3 !__
2 feet. What is the length of each side?
Let a " side length and c " diagonal length.
a !__
2 " c
a !__
2 " 3 !__
2
a " 3
The length of each side is 3 feet.
12. The length of the diagonal of a square is 80 millimeters. What is the length of each side?
Let a " side length and c " diagonal length.
a !__
2 " c
a !__
2 " 80
a " 80 ___ !
__ 2 " 80 ___
!__
2 # !
__ 2 ___
!__
2 " 80 !
__ 2 _____ 2 " 40 !
__ 2
The length of each side is 40 !__
2 millimeters.
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Calculate the area of each triangle.
13.
5 feet
45°
45°
14.
12.4 meters
45°
A " 1 __ 2 bh A " 1 __ 2 bh
A " 1 __ 2 (5)(5) A " 1 __ 2 (12.4)(12.4)
A " 1 __ 2 (25) A " 1 __ 2 (153.76)
A " 12.5 square feet A " 76.88 square meters
15.
15 feet
45° 16.
4.8 inches
45°
45°
(side length) !__
2 " 15
side length " 15 ___ !
__ 2 " 15 ___
!__
2 # !
__ 2 ___
!__
2 " 15 !
__ 2 _____ 2 feet
A " 1 __ 2 bh
A " 1 __ 2 ( 15 !__
2 _____ 2 ) ( 15 !__
2 _____ 2 ) A " (15 !
__ 2 )(15 !
__ 2 ) ____________
(2)(2)(2)
A " (15)(15) _______ (2)(2)
" 225 ____ 4 " 56.25
A " 56.25 square feet
(side length) !__
2 " 4.8
side length " 4.8 ___ !
__ 2 " 4.8 ___
!__
2 # !
__ 2 ___
!__
2
" 4.8 !__
2 ______ 2 " 2.4 !__
2 inches
A " 1 __ 2 bh
A " 1 __ 2 (2.4 !__
2 )(2.4 !__
2 )
A " (2.4 !__
2 )(2.4 !__
2 ) _____________ (2)
A " (2.4)(2.4)(2) ___________ (2)
" (2.4)(2.4) " 5.76
A " 5.76 square inches
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Calculate the area of each square.
17.
6.2 meters
18.
12 inches
A " s2 A " s2
A " (6.2)2 A " (12)2
A " 38.44 square meters A " 144 square inches
19.
5 cm
20.
17 mm
(side length) !__
2 " (diagonal length)
s !__
2 " 5
s " 5 ___ !
__ 2 " 5 ___
!__
2 # !
__ 2 ___
!__
2 " 5 !
__ 2 ____
2
A " s2
A " ( 5 !__
2 ____ 2 ) 2 A " ( 5 !
__ 2 ____ 2 ) ( 5 !
__ 2 ____ 2 )
A " 25 ___ 2 " 12.5
A " 12.5 square centimeters
(side length) !__
2 " (diagonal length)
!__
2 " 17
s " 17 ___ !
__ 2 " 17 ___
!__
2 # !
__ 2 ___
!__
2 " 17 !
__ 2 _____ 2
A " s2
A " ( 17 !__
2 _____ 2 ) 2 A " ( 17 !
__ 2 _____ 2 ) ( 17 !
__ 2 _____ 2 )
A " 289 ____ 2 " 144.5
A " 144.5 square millimeters
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Skills Practice Skills Practice for Lesson 3.4
Name _____________________________________________ Date_____________________
Finishing ConcreteProperties of 30º!60º!90º Triangles
Vocabulary Answer each question in your own words.
1. How are the side lengths of a 30º!60º!90º triangle related?
In a 30º!60º!90º triangle with a shorter leg of length a, the hypotenuse has a length of 2a. The longer leg has a length of a !
__ 3 .
2. How is a 30º!60º!90º triangle used to find the altitude of an equilateral triangle?
When an altitude is drawn in an equilateral triangle, it forms two congruent 30º!60º!90º triangles. The side length of the equilateral triangle becomes the length of the hypotenuse in the 30º!60º!90º triangle. The 30º!60º!90º triangle side length relationship can be used to find the altitude. The altitude is equal to 1 __ 2 (side length)( !
__ 3 ).
Problem Set Use the Pythagorean Theorem to calculate the missing side length of each triangle. Leave radicals in simplest form.
1. 2.
c
8!3
8 a
7!3 cm
14
a " 8, b " 8 !__
3 b " 7 !__
3 , c " 14
a2 # b2 " c2 a2 # b2 " c2
(8)2 # (8 !__
3 )2 " c2 a 2 # (7 !__
3 )2 " (14)2
64 # 64(3) " c2 a2 " (14)2 ! (7 !__
3 )2
256 " c2 a2 " 196 ! 147
16 " c a2 " 49
c " 16 units a " 7 centimeters
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3. A right triangle has a hypotenuse length of 40 units and one leg length of 20 units. Find the length of the other leg.
Let a " 20, c " 40.
a2 # b2 " c2
(20)2 # b2 " (40)2
400 # b2 " 1600
b2 " 1600 ! 400
b2 " 1200
b " !_____
1200
b " !____
400 !__
3
b " 20 !__
3
The other leg is 20 !__
3 units long.
4. A right triangle has a hypotenuse length of 12 units and one leg length of 6 units. Find the length of the other leg.
Let a " 6, c " 12.
a2 # b2 " c2
(6)2 # b2 " (12)2
36 # b2 " 144
b2 " 144 ! 36
b2 " 108
b " !____
108
b " !___
36 !__
3
b " 6 !__
3
The other leg is 6 !__
3 units long.
Chapter 3 l Skills Practice 369
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Use the 30º!60º!90º Triangle Theorem to calculate the missing side lengths in each triangle. Leave radicals in simplest form.
5. 6.
b
a
30°
4 cm60°
b
a
30°
18 m
60°
shorter leg: 2a " 4 shorter leg: 2a " 18
a " 2 cm a " 9 m
longer leg: b " a !__
3 " 2 !__
3 cm longer leg: b " a !__
3 " 9 !__
3 m
7. 8.
c
a
30°
60°
!6 ft
c
a
30°
3!2 in
60°
shorter leg: a !__
3 " b shorter leg: a !__
3 " b
a !__
3 " !__
6 a !__
3 " 3 !__
2
a " !__
6 ___ !
__ 3 " ( !
__ 6 ___
!__
3 ) ( !
__ 3 ___
!__
3 ) "
!________
(2)(3)(3) _________ 3 " 3 !
__ 2 ____ 3 " !
__ 2 ft a " 3 !
__ 2 ____
!__
3 " ( 3 !
__ 2 ____
!__
3 ) ( !
__ 3 ___
!__
3 ) " 3 !
__ 6 ____ 3 " !
__ 6 in
hypotenuse: c " 2a hypotenuse: c " 2a
c " 2 !__
2 ft c " 2 !__
6 in
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9. A 30º!60º!90º triangle has a shorter leg that measures 30 inches. Find the length of the longer leg and the length of the hypotenuse.
a " 30
longer leg: b " a !__
3 " 30 !__
3 inches
hypotenuse: c " 2a
c " 2(30) " 60 inches
10. A 30º!60º!90º triangle has a shorter leg that measures 14 inches. Find the length of the longer leg and the length of the hypotenuse.
a " 14
longer leg: b " a !__
3 " 14 !__
3 inches
hypotenuse: c " 2a
c " 2(14) " 28 inches
Chapter 3 l Skills Practice 371
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Use the 30º!60º!90º Triangle Theorem to calculate the missing measurement of each equilateral triangle. Leave radicals in simplest form.
11. An equilateral triangle has a side length of 100 inches. What is the measurement of the altitude?
The altitude divides the equilateral triangle into two 30º!60º!90º triangles.
The side length is the hypotenuse and the altitude is the longer leg.
Find the shorter leg first.
hypotenuse: c " 2a
100 " 2a
a " 50
longer leg: b " a !__
3 " 50 !__
3 inches
The measurement of the altitude is 50 !__
3 inches.
12. An equilateral triangle has a side length of 22 inches. What is the measurement of the altitude?
The altitude divides the equilateral triangle into two 30º!60º!90º triangles.
The side length is the hypotenuse and the altitude is the longer leg.
Find the shorter leg first.
hypotenuse: c " 2a
22 " 2a
a " 11
longer leg: b " a !__
3 " 11 !__
3 inches
The measurement of the altitude is 11 !__
3 inches.
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13. The altitude of an equilateral triangle has a measurement of 4 !__
3 feet. What is the side length?
The altitude divides the equilateral triangle into two 30º!60º!90º triangles.
The side length is the hypotenuse and the altitude is the longer leg.
Find the shorter leg first.
shorter leg: a !__
3 " b
a !__
3 " 4 !__
3
a " 4
hypotenuse: c " 2a
c " 2(4) " 8
The side length is 8 feet.
14. The altitude of an equilateral triangle has a measurement of 42 !__
3 millimeters. What is the side length?
The altitude divides the equilateral triangle into two 30º!60º!90º triangles.
The side length is the hypotenuse and the altitude is the longer leg.
Find the shorter leg first.
shorter leg: a !__
3 " b
a !__
3 " 42 !__
3
a " 42
hypotenuse: c " 2a
c " 2(42) " 84
The side length is 84 millimeters.
Chapter 3 l Skills Practice 373
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Calculate the area of each triangle.
15. 16.
30°
8 in60°
30°
7 ft
60°
height " 4 inches height " 7 feet
base " 4 !__
3 inches base " 7 !__
3 feet
area " 1 __ 2 bh area " 1 __ 2 bh
" 1 __ 2 (4 !__
3 )(4) " 1 __ 2 (7 !__
3 )(7)
" 8 !__
3 square inches " 1 __ 2 (49 !__
3 )
" 49 !__
3 _____ 2 square feet
17. 18.
3 cm
60°
60°60°
70 mm
60°
60°60°
Draw an altitude. This is the height. Draw an altitude. This is the height.
base " 3 centimeters base " 70 millimeters
height " 1 __ 2 (3) !__
3 height " 1 __ 2 (70) !__
3
" 3 !__
3 ____ 2 " 35 !__
3
area " 1 __ 2 bh area " 1 __ 2 bh
" 1 __ 2 (3) ( 3 !__
3 ____ 2 ) " 1 __ 2 (70)(35 !__
3 )
" (3)(3 !__
3 ) ________ (2)(2)
" 1225 !__
3 square millimeters
" 9 !__
3 ____ 4
" 9 !__
3 ____ 4 square centimeters
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Calculate the surface area and volume of each triangular prism. Round decimals to the nearest tenth.
19.
Triangular base:
a " 2, b " 2 !__
3 , c " 4
B " 1 __ 2 (2)(2 !__
3 ) " 2 !__
3
P " 2 # 2 !__
3 # 4 " 6 # 2 !__
3
S " 2(2 !__
3 ) # (6 # 2 !__
3 )(12)
" 4 !__
3 # 72 # 24 !__
3
" 72 # 28 !__
3
! 120.5 square inches
V " 1 __ 2 (2)(2 !__
3 )(12)
" 24 !__
3
! 41.6 cubic inches
20.
Half of triangular base:
a " 3, b " 3 !__
3 , c " 6
B " 1 __ 2 (6)(3 !__
3 ) " 9 !__
3
P " 6 # 6 # 6 " 18
S " 2(9 !__
3 ) # (18)(20)
" 18 !__
3 # 360
! 391.2 square centimeters
V " 1 __ 2 (6)(3 !__
3 )(20)
" 9 !__
3 (20)
" 180 !__
3
! 311.8 cubic centimeters
30°
2 in
60°
12 in
6 cm
60°60°20 cm
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21.
30° 30°
15 ft
18 ft a " 7.5, b " 7.5 !__
3 , c " 15
B " 1 __ 2 (15)(7.5 !__
3 ) " 56.25 !__
3
P " 15 # 15 # 15 " 45
S " 2(56.25 !__
3 ) # 45(18)
" 112.5 !__
3 # 810
! 1004.9 square feet
V " 1 __ 2 (15)(7.5 !__
3 )(18)
" 1012.5 !__
3
! 1753.7 cubic feet
22.
60°
32 m
9 m
a " 9, b " 9 !__
3 , c " 18
B " 1 __ 2 (9)(9 !__
3 ) " 40.5 !__
3
P " 9 # 9 !__
3 # 18 " 27 # 9 !__
3
S " 2(40.5 !__
3 ) # (27 # 9 !__
3 )(32)
" 81 !__
3 # 864 # 288 !__
3
" 369 !__
3 # 864
! 1503.1 square meters
V " 1 __ 2 (9)(9 !__
3 )(32)
" 1296 !__
3
! 2244.7 cubic meters
