Name: ________________________ Class: ___________________ Date: __________
ID: A
Unit 4 Practice Packet
____
3
7
= , then x = ______.
x −4 x
7
a.
b. 7
3
1. If
c.
4
d.
3
____
2. Mr. Jones has taken a survey of college students and found that 1 out of 4 students are liberal arts majors. If a
college has 7000 students, what is the best estimate of the number of students who are liberal arts majors?
a. 28,000
b. 140
c. 175
d. 1750
____
3. A worker in an assembly line takes 9 hours to produce 29 parts. At that rate, how many parts can she produce
in 45 hours?
a. 4 parts
b. 145 parts
c. 290 parts
d. 1305 parts
____
4. While attending a school carnival, you estimate the ratio of children to adults as 2: 1. If there are 285 people
at the carnival, about how many children are in attendance?
a. about 238
b. about 190
c. about 285
d. about 95
5. The ratios of the side lengths of triangle ABC are 7:9:12 (AB:AC:BC). Solve for x.
Solve:
6.
11
x
=
26 15
7.
Solve the proportion
5
7
= .
x−1 x
8.
Solve the proportion
3
7
= .
2x
5
9. The official width-to-length ratio of the United States flag is 1:1.9. If a United States flag is 9.5 feet long,
how wide should it be?
10. A board 18 inches long is cut into two pieces in the ratio 1: 5. Find the length of each piece.
11. The measures of the angles of a triangle are in the extended ratio of 7 : 9 : 10. Find the measures of the angles
of the triangle.
12. A triangle with a perimeter of 63 feet has side lengths in the extended ratio of 6 : 7 : 8. Find the side lengths
of the triangle.
1
Name: ________________________
____ 13. Given that
a.
a.
ED EC
=
, find BC to the nearest tenth. The figure is not drawn to scale.
BA BC
8.9
____ 14. Given that
ID: A
b.
19.1
c.
21.3
d.
5.1
ED EC
=
, find AB to the nearest tenth. The figure is not drawn to scale.
BA BC
16.0
b.
18.2
c.
17.1
d.
14.9
1
inch : 10 miles. If the actual distance between the two cities is 340 miles, how far
2
apart are they on the map?
a. 17 inches
b. 68 inches
c. 8.5 inches
d. 34 inches
____ 15. A map has a scale of
16. The product label glued to a box of fruit is 10 inches wide by 9 inches tall. Part of the company logo on the
label is a circle that has a diameter of 6 inches. An enlarged copy of the label, 20 inches by 18 inches, is used
on a larger box. What is the circumference of the circle in the enlarged company logo? Use 3.14 as an
approximation for π and round your answer to the nearest tenth of an inch.
____ 17. If two polygons are SIMILAR, then the corresponding angles must be _____.
a. complementary
c. congruent
b. supplementary
d. linear pairs
____ 18. If two polygons are SIMILAR, then the corresponding sides must be _____.
a. proportional
c. parallel
b. congruent
d. similar
____ 19. ∆ABC and ∆XYZ are similar with ∠A = ∠X, and ∠B = ∠Y. If AB, BC, and AC are 7 inches, 9 inches, and 10
inches, respectively, and XY is 9 inches, find XZ.
a. 7 in.
b. 7.8 in.
c. 11.6 in.
d. 12.9 in.
2
Name: ________________________
ID: A
____ 20. Given that ∆ABC ∼ ∆DEF, solve for x and y.
a.
b.
x = 10.67, y = 13.5
x = 9.67, y = 13.5
c.
d.
x = 10.67, y = 12.5
x = 9.67, y = 12.5
____ 21. The perimeter of ∆PQR is 80, PQ = 30, ∆PQR ∼∆STU, and ST = 18. What is the perimeter of ∆STU?
a. 18.4
b. 6.8
c. 48
d. 24
22. A rectangle has length 15 cm. Another rectangle is drawn using a scale factor of
2
. What is the length of the
3
second rectangle?
23. A photo needs to be enlarged from an original with a length of 12 inches and a width of 10 inches to a size
where the new width is 50 inches. What is the new length? What is the scale factor?
____ 24. Which triangle is NOT similar to any of the others?
b.
a.
c.
d.
____ 25. Two ladders are leaning against a wall at the same angle as shown. How long is the shorter ladder?
a.
18 ft
b.
22 ft
c.
3
36 ft
d.
8 ft
Name: ________________________
ID: A
____ 26. Triangles LMN and NWR are right triangles.
What is the length of NW?
a. 2.5 cm
b. 10 cm
c.
15.6 cm
d.
14.4 cm
____ 27. Marcia wants to measure the height of the flagpole at her school. She places a mirror on the ground 56 feet
from the flagpole, then walks backward until she is able to see the top of the flagpole in the mirror. Her eyes
are 5.5 feet above the ground, and she is 11 feet from the mirror. What is the height of the flagpole to the
nearest tenth of a foot?
a. 112.0
b. 28.0
c. 29.1
d. 12.0
28. Melody wants to find the height of the tallest building in his city. He stands 422 feet away from the building.
There is a tree 40 feet in front of him, which he knows is 22 feet tall. How tall is the building? (Round to the
nearest foot.)
Determine whether the triangles are similar. If the are, write a similarity statement.
29.
30.
4
Name: ________________________
ID: A
31. Lena wants to make ∆JKL similar to ∆ABC. The points J and K have been plotted on the graph below.
Where can Lena plot point L on the graph to make ∆JKL similar to ∆ABC? Explain your answer.
Draw the image of the given figure after a dilation with center O and the given scale factor.
32. scale factor:
1
2
33. scale factor: 2
5
Name: ________________________
ID: A
34. Quadrilateral A′B ′C ′D ′ is the image of quadrilateral ABCD after a dilation with a scale factor of 3. Using
the origin as the center of the dilation, draw A′B ′C ′D ′ and label its vertices with their coordinates.
Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement.
35.
36.
____ 37. Find the length of the leg of this right triangle. Give an approximation to 3 decimal places.
a.
b.
12.329
11.916
c.
d.
6
12.650
27.019
Name: ________________________
ID: A
____ 38. How long is a string reaching from the top of a 16-ft pole to a point on the ground that is 5 ft from the base of
the pole?
291 ft
c.
281 ft
a.
b.
241 ft
d.
231 ft
____ 39. ∆ABC is a right triangle. AB = _____.
a.
b.
3 13
3 6
c.
d.
3 5
117
____ 40. The city commission wants to construct a new street that connects Main Street and North Boulevard as
shown in the diagram below. The construction cost has been estimated at $100 per linear foot. Find the
estimated cost for constructing the street. (1 mile = 5280 ft)
a.
b.
$528,000
$47,226
c.
d.
7
$4,767,020
$4,722,576
Name: ________________________
ID: A
____ 41. A 25.5 foot ladder rests against the side of a house at a point 24.1 feet above the ground. The foot of the
ladder is x feet from the house. Find the value of x to one decimal place.
a.
1.9
b.
7.0
c.
8.3
d.
10.1
____ 42. A scuba diver has a taut rope connecting the dive boat to an anchor on the ocean floor. The rope is 90 feet
long. The water is 40 feet deep. To the nearest tenth of a foot, how far is the anchor from a point directly
below the boat?
a. 80.6 ft
c. 71.7 ft
b. 75.1 ft
d. 98.5 ft
43. Find the altitude of an isosceles triangle with base 10 and congruent sides of length 9.
44. A power pole broke and fell as shown.
To the nearest tenth of a meter, what was the original height of the pole?
8
Name: ________________________
ID: A
45. Find the area of this right triangle if b = 12 and c = 20.
46. A boat traveled in a straight line through calm seas until it was 43 kilometers west and 41 kilometers south of
its original position. Find how far the boat traveled, to the nearest tenth of a kilometer.
____ 47. In a 45°-45°-90° triangle, the ratio of the length of the hypotenuse to the length of a side is _____.
3 :1
c.
2 :1
d. 2:1
a. 1:1
b.
____ 48. The shorter leg of a 30°-60°-90° triangle is 8.5 feet long. Find the perimeter.
Ê
ˆ
Ê
ˆ
a. ÁÁÁ 25.5 + 8.5 2 ˜˜˜ ft
c. ÁÁÁ 17 + 8.5 2 ˜˜˜ ft
Ë
¯
Ë
¯
ÊÁ
ˆ˜
ÊÁ
ˆ
b. ÁÁ 17 + 8.5 3 ˜˜ ft
d. ÁÁ 25.5 + 8.5 3 ˜˜˜ ft
Ë
¯
Ë
¯
____ 49. An equilateral triangle has side lengths of 10. The length of its altitude is _____.
a. 10 5
b. 5
c. 5 10
d. 5 3
____ 50. In a 30°-60°-90° triangle, the ratio of the length of the hypotenuse to the length of the shorter side is _____.
a. 2: 3
b.
2 :1
c. 2:1
d.
3 :1
51. Find the value of x and y.
52. What is the length of the diagonal of a square with side lengths 7 2 ?
9
Name: ________________________
ID: A
53. Find the value of x and y.
54. Find the value of x and y.
____ 55. A photographer shines a camera light at a particular painting forming an angle of 40° with the camera
platform. If the light is 58 feet from the wall where the painting hangs, how high above the platform is the
painting?
a.
1.19 ft
b.
48.67 ft
c.
69.12 ft
d.
0.84 ft
56. Find tan A for the right triangle below:
____ 57. A slide 2.5 m long makes an angle of 26° with the ground. How high is the top of the slide above the ground?
a. 1.22 m
b. 1.13 m
c. 2.25 m
d. 1.1 m
____ 58. Liola drives 16 km up a hill that is at a grade of 10o . What horizontal distance, to the nearest tenth of
kilometer, has she covered?
a. 14.7 km
b. 15.3 km
c. 15.8 km
d. 17.1 km
10
Name: ________________________
ID: A
59. A tree 19 feet tall casts a shadow which forms an angle of 49° with the ground. How long is the shadow to
the nearest hundredth?
60. a. What is a tangent ratio?
b. Find tan50 o , tan60 o , and tan70 o .
c. What happens to the tangent ratio as the size of the angle increases?
d. What is tan90o ? Explain why your answer is reasonable.
____ 61. Write cos A.
a.
15
8
b.
8
17
c.
15
17
62. Find the value of x, to the nearest whole number. (not drawn to scale)
11
d.
8
15
Name: ________________________
ID: A
____ 63. Use the diagram to find cos x as a fraction in simplest form.
a.
b.
12
13
2
2
5
c.
d.
5
12
5
13
64. A 220 ft string attached to a kite makes a 30o angle with the ground. What is the height of the kite to the
nearest tenth?
65. Solve ∆ABC using the diagram and the given measurements.
(Note: The triangle is not drawn to scale.)
B = 49°, a = 4
____ 66. Assume that ∠A is an acute angle and tan A = 1.230. The measure of ∠A is _____.
a. about 39.1°
c. about 50.9°
b. about 7.01°
d. about 129.9°
67. A 30°-60°-90° triangle is shown below. Draw another triangle similar to the given triangle, indicating the
lengths of the sides. Show that the values of the sine, cosine, and tangent of the 30° and 60° angles of your
triangle are the same as those for the given triangle.
12
Name: ________________________
ID: A
68. Can you find AB using the sine ratio? Can you find AB using the cosine ratio? Explain.
____ 69. Find the missing angle and side measures of ∆ABC, given that m∠A = 65°, m∠C = 90°, and CB = 15.
a. m∠ B = 25°, c = 16.6, b = 7
b. m∠ B = 155°, c = 16.6, b = 7.5
c. m∠ B = 155°, c = 16.6, b = 7
d. m∠ B = 25°, c = 16.1, b = 7
____ 70. Solve for x to the nearest degree.
a.
30
b.
63
c.
60
d.
27
____ 71. Two legs of a right triangle have lengths 15 and 8. The measure of the smaller acute angle is _____.
b. ≈ 17°
c. ≈ 61.9°
d. ≈ 28.1°
a. ≈ 32.2°
72. An airplane is flying at an elevation of 1500 feet. What is the airplane's angle of elevation from the runway
when it is 5000 feet from the runway? Explain.
1
bh, where b is the length of the base of the triangle
2
and h is the height of the triangle. In order to use the formula to find the area of the triangle below, you must
first draw the altitude from the largest angle and then find the lengths of the needed sides. Find the area of the
triangle.
73. To find the area of a triangle, you use the formula A =
13
ID: A
Unit 4 Practice Packet
Answer Section
1.
2.
3.
4.
5.
B
D
B
B
6
9
26
7
7. x =
2
6. 6
8. x =
15
14
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
5 ft
3 in., 15 in.
35 û , 45 û , 100 û
18 feet, 21 feet, 24 feet
B
D
A
37.7 inches
C
A
D
A
C
10 cm
new length = 60 inches; scale factor = 5
D
A
C
B
232 ft
Similar; ∆EBC ∼ ∆EDA
Similar; ∆ABE ∼ ∆CDE
31. ÊÁË 2, − 1 ˆ˜¯ or ÊÁË 2, − 19 ˆ˜¯ ; Explanations may vary. Sample explanation: AB corresponds to JK . The ratio of JK
JK 6 3
3
JL
JL 3
to AB is
= = . The ratio of JL to AC must also equal . Since
=
= , the length JL must
AB 4 2
2
AC
6
2
be 9 units. Angle A and angle J are both right angles and JK is a horizontal line, so JL must be a vertical
line. The points ÁÊË 2, − 1 ˜ˆ¯ and ÁÊË 2, − 19 ˜ˆ¯ are both 9 vertical units away from point J.
1
ID: A
32.
33.
34.
35. 3; enlargement
1
; reduction
36.
4
37. A
38. C
39. A
40. D
41. C
42. A
43.
44.
45.
46.
47.
48.
49.
56 or 2 14
18.5
96
59.4 kilometers
C
D
D
2
ID: A
50. C
51. x = 11 2, y = 11 + 11 3 or 11(1 +
52. 14
3)
53. x = 3 3, y = 6
54. x = 13, y = 13 3
55. B
5
56.
12
57. D
58. C
59. 16.52 ft
60. a. In a right triangle, the tangent ratio of an acute angle is the ratio of the length of the leg opposite the angle
to the length of the leg adjacent to the angle.
b. tan 50° = 1.19, tan 60° = 1.73, and tan 70° = 2.75
c. As the size of the angle increases, the tangent ratio increases.
AE
. As m∠D approaches 90û , ED approaches 0. So,
d. Undefined. tanD =
ED
AE
tan90 o =
, and a fraction with 0 in the denominator is undefined.
0
61. C
62. 5
63. A
64. 110.0 ft
65. A = 41 o , b = 4.60, c = 6.10
66. C
67. Sample answer: For the given triangle, sin 30 o =
cos 60 o =
1
, and tan 60o =
2
3.
For the similar triangle below, sin 30 o =
sin 60 o =
3
3
1
1
, tan 30o =
,
, cos 30 o =
, sin 60 o =
2
2
2
3
5 3
3
5
1
5
1
=
, tan 30 o =
= , cos 30 o =
=
,
10 2
10
2
5 3
3
5 3
3
5 3
5
1
=
, cos 60 o =
=
= , and tan 60o =
10
2
10 2
5
3
3.
ID: A
68. You can use both the sine and cosine ratios to find AB. To find AB, use sinB =
AC
AC
or cos A =
. Since
AB
AB
10
10
means that AB =
. By the Triangle Sum Theorem, m∠A = 25 o .
o
AB
sin65
10
10
10
10
So, cos 25 o =
means that AB =
. You can use a calculator to see that
=
.
o
o
AB
cos 25
cos 25 o
sin65
A
A
D
ÊÁ 1500 ˆ˜
1500
˜˜˜ ≈ 72.5 o
About 72.5 o . cos x =
so x = cos −1 ÁÁÁÁ
˜
5000
5000
Ë
¯
m∠B = 65 o and AC = 10, sin65 o =
69.
70.
71.
72.
73. Drawing the altitude from the 105° angle divides the triangle into two smaller right triangles. The triangle on
the left is a 30 o − 60 o − 90 o triangle and the triangle on the right is a 45 o − 45 o − 90 o triangle. Since the length
of the hypotenuse of the 30 o − 60 o − 90 o triangle is 48 cm, the length of the altitude that was drawn is 24 cm
and the length of the other leg of the 30 o − 60 o − 90 o triangle is 24 3 cm. Since the length of the altitude is
24 cm, then the length of the other leg of the 45 o − 45 o − 90 o triangle is also 24 cm. Thus, the length of the
1Ê
ˆ
Ê
ˆ
base is 24 cm. Therefore, the area of the triangle is ÁÁÁ 24 3 + 24˜˜˜ (24), or ÁÁÁ 288 3 + 288 ˜˜˜ cm2 (this is
2Ë
¯
Ë
¯
2
about 786.8 cm ).
4