Download Geometry EOI Practice Trigonometric Ratios 1. If = 58° and m = 120

Geometry EOI Practice
Trigonometric Ratios
1.
If = 58° and m = 120 in, what is the value of p to the nearest tenth of an inch?
A. 108.3 in
B. 75 in
C. 141.5 in
D. 63.6 in
Pythagorean Theorem
2. If two sides of an obtuse triangle are 15 cm and 21 cm long, which of the following measurements could be the
length of the third side?
A. 21 cm
B. 27 cm
C. 24 cm
D. 18 cm
Properties of Figures
3. What is the most specific name for figure KLMN?
A. rhombus
B. kite
C. trapezoid
D. parallelogram
3-Dimensional Models
4. A two-dimensional net is shown below.
Which three-dimensional shape can be made with this net?
A. cylinder
B. sphere
C. cone
D. cube
Polygons and Other Plane Figures
5.
Figure not drawn to scale
If figure JKLM is an isosceles trapezoid with JP = 180 inches, KM = 468 inches, and KL = 204 inches, what is the
length of side LM?
A. 480 inches
B. 528 inches
C. 96 inches
D. 432 inches
Properties of Figures
6. The vertices of a triangle are listed below.
Which of the following correctly classifies the triangle?
A. The triangle is an acute equilateral triangle.
B. The triangle is a right scalene triangle.
C. The triangle is a right isosceles triangle.
D. The triangle is an acute scalene triangle.
Special Right Triangles
7. If x = 15 inches, what is the perimeter of the figure above?
A. (112.5
+ 225) inches
B. (15
+ 30
D. (30
+ 60) inches
C. (60
+ 30
+ 45) inches
+ 30) inches
Distance, Midpoint, and Slope
8.
A.
B.
C.
D.
Distance, Midpoint, and Slope
9. What is the distance between the points (3, 7) and (-9, -9) in the coordinate plane?
A. 12
B. 20
C. 16
D. 15
Line and Angle Relationships
10.
Which of the following are alternate exterior angles?
A. A and E
B.
A and
D.
C and
C.
B and
H
E
F
Circles
11. The center of the circle is R.
Arc length of
2
Figure not drawn to scale.
If the radius of
A.
24
R is 9 inches, what is the arc length of
in
B.
in
C.
in
D.
in
Distance, Midpoint, and Slope
12.
A.
B.
C.
D.
?
r
=
m
360
Polyhedra and Other Solids
13.
If L = 12 in, W = 3 in, and H = 4 in, what is the surface area of the rectangular prism?
A. 192 in2
B. 150 in2
C. 234 in2
D. 144 in2
Distance, Midpoint, and Slope
14. Line r has a slope of -11. Line s is parallel to line r. What is the slope of line s?
A.
B.
C.
D.
3-Dimensional Models
15.
Which is the fewest number of cubes needed to build the object shown above?
A. 32
B. 31
C. 34
D. 33
Line and Angle Relationships
16.
Lines RS and TU are parallel. If
O equals 45°, then what is the measure of
A. 90°
B. 135°
C. 45°
D. 37°
3-Dimensional Models
17. Which net can be folded to create a pentagonal prism?
B.
A.
C.
D.
N?
Special Right Triangles
18. The triangle-shaped clock below has measurements that are approximately equal to the measurements of an
equiangular triangle.
If one side of the triangle measures 198 mm, approximately how tall is the clock?
Picture is not drawn to scale.
A.
B.
C.
D.
Pythagorean Theorem
19. If two sides of an acute triangle are 18 cm and 22 cm long, which of the following measurements could be the
length of the third side?
A. 12 cm
B. 10 cm
C. 30 cm
D. 28 cm
Polyhedra and Other Solids
20. If the base of a prism has 6 sides, how many vertices does the prism have?
A. 18
B. 8
C. 12
D. 7
Properties of Figures
21. The vertices of a quadrilateral are listed below.
Q(-3 , 3), R(5 , 3), S(3 , -3), T(-5 , -3)
Which of the following is the strongest classification that identifies this quadrilateral?
A. The quadrilateral is a rhombus.
B. The quadrilateral is a parallelogram.
C. The quadrilateral is a square.
D. The quadrilateral is a rectangle.
Similarity and Congruence
22. The two pyramids below are similar.
20 cm
35 cm
If the front base edge of the top pyramid is 28 cm, what is the length of the front base edge of the bottom pyramid?
A. 56 cm
B. 43 cm
C. 49 cm
D. 25.00 cm
3-Dimensional Models
23.
Which of the following pictures represents a front view of this solid?
A.
B.
C.
D.
Pythagorean Theorem
24. Which of the following measurements could be the side lengths of a right triangle?
A. 12 in, 16 in, 24 in
B. 8 in, 16 in, 20 in
C. 12 in, 16 in, 20 in
D. 16 in, 20 in, 24 in
Properties of Figures
25. What are the coordinates of point R' if figure Q'R'S'T' is a reflection of figure QRST across the x-axis?
A. (2,7)
B. (7,2)
C. (-2,7)
D. (7,-2)
Line and Angle Relationships
26. Lines RS and TU are parallel. If the measure of
X equals 69°, then what is the measure of
O?
A. 69°
B. 66°
C. 21°
D. 111°
Polygons and Other Plane Figures
27. Which of the following polygons appears to be a regular polygon?
A.
C.
D.
B.
Special Right Triangles
28. The shorter leg of a 30°-60°-90° triangle measures
A.
B.
C.
D.
inches
inches
inches
inches
inches. What is the length of the longer leg?
Polyhedra and Other Solids
29.
How many edges does a rectangular pyramid have?
A. 12
B. 6
C. 5
D. 8
Special Right Triangles
30. One leg of a 45°-45°-90° triangle measures 7 inches. What is the length of the hypotenuse?
A. inches
B.
C.
D.
inches
inches
inches
Polyhedra and Other Solids
31. Mr. Wilson is a geography teacher. He has 5 globes in his classroom. Each globe has a diameter of 18 inches. What
is the surface area of all of the globes in his classroom?
A. 15,260.40 in2
B. 1,017.36 in2
C. 5,086.8 in2
D. 3,052.08 in2
Polygons and Other Plane Figures
32. Choose the correct word to complete the sentence.
A. hexagon
B. quadrilateral
C. pentagon
D. octagon
A polygon with five vertices is a(n) ____________.
Congruence
33.
Given: The length of AB is 48 units.
The length of BC is 32 units.
The length of AC is 58 units.
If
ABC is congruent to
A. 32 units
XYZ, what is the length of XZ?
B. 64 units
C. 58 units
D. 48 units
Special Right Triangles
34. The hypotenuse of a 45°-45°-90° triangle measures 6 inches. What is the length of each leg?
A. inches
B.
C.
inches
D.
inches
inches
Circles
35.
Figure is not drawn to scale.
If m FDG = 30°, what is m
A. 30°
B. 15°
C. 45°
D. 60°
FEG?
Congruence
36. Figure QRST and figure WXYZ are congruent parallelograms.
Figure is not drawn to scale.
What is the m W?
A. 75°
B. 73°
C. 90°
D. 107°
Similarity and Congruence
37. Cone E has a height of 60 cm. The base of cone F has an area of 625
congruent, what is the surface area of cone E?
A. 2,250
cm2
C. 1,625
cm2
B. 12,500
D. 2,125
A=
SA =
r2
rl +
cubic centimeters. If cones E and F are
r2
cm2
cm2
Similarity and Congruence
38. Jason has two congruent boxes of snacks: a box of crackers and a box of cookies.
*Note: Picture not drawn to scale.
The area of the square base of the box of crackers is 25 in 2. The volume of the box of cookies is 285 in3.
What is the length, x, of the box of crackers?
A. 5 in
B. 16.4 in
C. 11.4 in
D. 10 in
Special Right Triangles
39.
What is m
A. 45°
MNO?
B. 60°
C. 30°
D. 90°
Similarity
40. Ben formed a triangle using three pencils, shown below.
He then broke each pencil exactly in half and formed a new triangle with three of the pencil pieces. With the
information given, determine how the new triangle and the original triangle can be shown to be similar.
A. The triangles are not similar to each other.
B. The triangles are similar by SSS.
C. The triangles are similar by AA.
D. The triangles are similar by SAS.
Pythagorean Theorem
41.
If S = 7 cm and T = 24 cm, what is the length of R?
A.
B.
C.
D.
Polygons and Other Plane Figures
42. In kite PQRS, PR = 16 cm and QS = 21 cm.
If the ratio of QT to TS is 2 to 5, then what is the perimeter of the kite?
A. 37 cm
B. 54 cm
C. 168 cm
D. 74 cm
Similarity and Congruence
43. Rectangular prisms A and B are congruent. Rectangular prism A has a length of 9 cm and a width of 3 cm. If
rectangular prism A has a volume of 324 cubic centimeters, what is the height of rectangular prism B?
A. 27 cm
V = lwh
B. 297 cm
C. 12 cm
D. 15 cm
Polygons and Other Plane Figures
44.
In rhombus ABCD, segments AC and BD are diagonals. If m DAB = 132°, what is the measure of
A. 156°
B. 24°
C. 204°
D. 114°
Polygons and Other Plane Figures
45.
Note: Figure not drawn to scale.
If m R = 2x2 - 7°, m S = 4x2 - 6x + 40°, and m T = 18x + 3°, what is m Q?
A. 155°
B. 137°
C. 169°
D. 115°
ABE?
Congruence
46. Lisa drew the following triangles to show that
congruence by the ASA theorem?
ABC
NLM by the ASA theorem. Does Lisa's drawing prove
A. No, the congruent sides are not included between the congruent angles.
B. No, Lisa did not mark enough congruent sides.
C. Yes, the triangles have a congruent side.
D. Yes, Lisa's drawing shows that one side and two angles of each triangle are congruent.
Circles
47. In the figure above, line TQ is tangent to a circle with center S at point Q and chord RQ is a diameter of the circle.
If x = 27°, what is m
PQT?
A. 72°
B. 63°
C. 54°
D. 117°
Congruence
48. In the diagram,
LMN
Which of these must be true?
A. LM ST
B. MN
C. LN
D. LN
US
UT
US
UST.
Note: not drawn to scale
Pythagorean Theorem
49.
Tri-Star Industries employs electrical, plumbing, and air conditioning
technicians. Their home office is made up of three square buildings, one for each
department, with a triangular atrium in the middle. If the area of the Plumbing
building is 3,600 ft2 and the area of the A/C building is 625 ft2, what is the area
of the Electrical building?
A. 3,025 ft2
B. 4,225 ft2
C. 2,975 ft2
D. 4,900 ft2
Trigonometric Ratios
50. A lamp illuminates an area that is 12 feet diagonally at an angle of 40° from the ground to the top of the lamp.
Which equation can be used to find the horizontal length, r, of the illuminated area?
A.
B.
C.
D.
Distance, Midpoint, and Slope
51.
A.
B.
C.
D.
Congruence
52. Fred drew the following triangles to show that
congruence by the AAS corollary?
A.
B.
C.
D.
ABC
LMN by the AAS corollary. Does Fred's drawing prove
Yes, Fred's drawing shows that one side and two angles of each triangle are congruent.
No, Fred marked the included sides congruent.
Yes, one side of the two triangles is congruent.
No, Fred did not mark enough congruent sides.
Distance, Midpoint, and Slope
53. Line p has an undefined slope. Line q is parallel to line p. What is the slope of line q?
A. Line q has an undefined slope.
B. 0
C. -1
D. 1
Trigonometric Ratios
54.
Note: Picture is not drawn to scale.
If X = 12 inches, Y = 16 inches, and Z = 20 inches, what is the tangent of
A.
B.
C.
D.
A?
Trigonometric Ratios
55.
If
= 27° and x = 9 cm, what is the value of z to the nearest tenth of a centimeter?
A. 10.1 cm
B. 19.8 cm
C. 17.7 cm
D. 18.8 cm
Pythagorean Theorem
56. Two cars leave a dealership at the same time. Car J is heading south on Interstate 45, and Car K is heading west on
Interstate 20. Two hours later they are 105 miles apart. If Car J had traveled 84 miles from the dealership, how many
miles had Car K traveled?
A. 42
B. 63
C. 84
D. 126
Properties of Figures
57. The vertices of a triangle are listed below.
Which of the following correctly classifies the triangle?
A. The triangle is a right scalene triangle.
B. The triangle is an obtuse isosceles triangle.
C. The triangle is an acute isosceles triangle.
D. The triangle is an acute equilateral triangle.
3-Dimensional Models
58.
Which of the following set of pictures represents a side and the bottom view of this solid?
A.
B.
C.
D.
Similarity and Congruence
59. Two similar rectangular prisms have lengths of 9 cm and 18 cm. If the base of the smaller prism has a width of 14
cm, what is the area of the base of the larger prism?
A. 252 cm2
B. 504 cm2
C. 46 cm2
D. 454 cm2
Polygons and Other Plane Figures
60.
Given the following measurements, what is the area of trapezoid ABCE?
A.
B.
C.
D.
Trigonometric Ratios
61. In the above triangle, if x = 1, y = 1, and z =
, then which of the following is equal to sin(45°)?
A.
B.
C.
D.
Circles
62. A circle has a radius of 6 inches and a central angle with a measure of 180°. What is the measure of the arc length
associated with this angle?
A. 12
inches
B. 1,080
C. 6
inches
inches
D. 1,080 inches
Line and Angle Relationships
63.
Which of the following are vertical angles?
A. A and E
B.
B and
E
D.
A and
B
C.
B and
C
3-Dimensional Models
64.
The pictures above represent the front, side, and top views of which of the objects below?
A. Z
W.
X.
Y.
B. W
C. X
D. Y
Polyhedra and Other Solids
65.
What is the sum of the number of faces, edges, and vertices for a hexagonal prism?
A. 32
B. 26
C. 38
D. 44
Z.
3-Dimensional Models
66. A 2-dimensional net is shown below.
Which of the following 3-dimensional shapes can be formed with this net?
A. triangular pyramid
B. triangular prism
C. rectangular pyramid
D. rectangular prism
3-Dimensional Models
67. The top, side, and front views of an object built with cubes are shown below.
How many cubes are needed to construct this object?
A. 10
B. 13
C. 17
D. 7
Similarity
69. In the figure below, the measure of
shown to be similar.
A. The triangles are similar by SSS.
m is 49° and the measure of
B. The triangles are not similar to each other.
C. The triangles are similar by AA.
D. The triangles are similar by SAS.
q is 41°. Determine how the triangles can be
Line and Angle Relationships
70. Which of the following are alternate interior angles?
A.
A and
B.
D and
D.
D and
C.
F and
D
F
G
E
Trigonometric Ratios
71. If X = 10 feet, Y = 24 feet, and Z = 26 feet, what is the cosine of
B?
A.
B.
C.
D.
Congruence
72. If H
Side rule?
K, which of the following conditions must also be met to prove that
A. I or IV only
I. HJ KJ
II. HJ LJ
III. GH LK
IV. GJ LJ
B. I or III only
C. II only
D. III or IV only
Polyhedra and Other Solids
73. If the base of a prism has 6 sides, how many edges does the prism have?
A. 12
B. 7
C. 18
D. 8
GHJ
LKJ by Angle-Angle-
Pythagorean Theorem
74. Firefighters have a 39-foot extension ladder. In order to reach 36 feet up a building, how far away from the building
should the foot of the ladder be placed?
A. 9 feet
B. 24 feet
C. 12 feet
D. 15 feet
Similarity
75. If triangles JLK and PRQ are similar, which of these equations must be true?
A.
B.
C.
D.
Similarity and Congruence
76. Michelle and Patrick went camping. They have matching, congruent cube-shaped coolers. The surface area of
Michelle's cooler is 486 square inches. What is the length of a side of Patrick's cooler?
A. 54 inches
SA = 6B
B. 18 inches
C. 9 inches
D. 81 inches
Special Right Triangles
77. A baseball field is a square with a side length of 90 ft. Approximately how far does a player throw the ball from
third base to first base?
A.
B.
C.
D.
Line and Angle Relationships
78.
Lines RS and TU are parallel. If
N equals 54°, then what is the measure of
Y?
A. 126°
B. 144°
C. 54°
D. 36°
Properties of Figures
79. The vertices of a quadrilateral are listed below.
W(8 , 6), X(11 , 3), Y(2 , -6), Z(-1 , -3)
Which of the following is the strongest classification that identifies this quadrilateral?
A. The quadrilateral is a square.
B. The quadrilateral is a trapezoid.
C. The quadrilateral is a rectangle.
D. The quadrilateral is a rhombus.
Polyhedra and Other Solids
80.
If the diameter of the sphere is 18 cm, what is the approximate surface area of the sphere?
A. 254.47 cm2
B. 1,017.88 cm2
C. 9,160.88 cm2
D. 508.94 cm2
Surface Area of a Sphere = 4
r2
Properties of Figures
81. If figure QRST is translated 6 units to the right and then translated 9 units down, which of the following will be
the coordinates for point S'?
A. (-7,12)
B. (5,-6)
C. (-10,9)
D. (5,3)
Pythagorean Theorem
82.
If X = 17 cm and Y = 8 cm, what is the length of Z?
A.
B.
C.
D.
Circles
83. The center of the circle is Q.
What is the measure of
A. 210°
?
B. 75°
C. 105°
D. 255°
Similarity
84. With the information given below, determine how triangle ACD can be shown to be similar to triangle ABE.
Point E is the midpoint of AD and point B is the midpoint of AC.
A. The triangles are similar by SAS.
Note: Picture not drawn to scale.
B. The triangles are not similar to each other.
C. The triangles are similar by SSS.
D. The triangles are similar by AA.
Polyhedra and Other Solids
85. The volume of the rectangular prism above is 540 in 3.
If L = 18 in and W = 5 in, what is the length of H?
A. 5 in
B. 6 in
C. 8 in
D. 7 in
Circles
86.
m BCD = 1/2(m
-m
)
If m
= 47° and m BCD = 29°, what is m
A. 134°
?
B. 105°
C. 47°
D. 76°
Polygons and Other Plane Figures
87. Brandon has designed the following logo for his bike shop. The right triangle is 12 cm tall and 9 cm wide.
Note: picture not drawn to scale
He hired a company to paint the logo on the bike shop wall 15 times bigger. Once it has been painted, Brandon will
outline the logo on the wall with some bicycle chain.
How long will the bike chain need to be to outline the logo painted on the bike shop wall?
A. 315 cm
B. 540 cm
C. 630 cm
D. 585 cm
Trigonometric Ratios
88. Daniel sees a lighthouse in the harbor. He estimates the angle of elevation is 50°. If the lighthouse is 140 feet tall,
what is the approximate distance between Daniel and the top of the lighthouse? (Assume the lighthouse meets the
ground at a right angle.)
A. 183 feet
B. 117 feet
C. 107 feet
D. 218 feet
Congruence
89. Circle P and circle S are congruent. The area of circle P is 50.24 square inches. To the nearest tenth of an inch, what
is the circumference of circle S? (Use 3.14 for .)
A. 25.1 inches
B. 12.6 inches
C. 3.1 inches
D. 100.5 inches
Properties of Figures
90.
What is the rule for the transformation shown above?
A. (x' , y') = (-y , x)
B. (x' , y') = (-x , -y)
C. (x' , y') = (-y , -x)
D. (x' , y') = (x , -y)
A = r2
C=2 r
Special Right Triangles
91. The shorter leg of a 30°-60°-90° triangle measures
inches
A.
inches
B.
C.
D.
inches. What is the length of the hypotenuse?
inches
inches
Line and Angle Relationships
92.
Lines RS and TU are parallel. If the measure of
O equals 65°, what is the measure of
A. 160°
B. 65°
C. 70°
D. 115°
Circles
93. The center of the circle is Q.
Figure not drawn to scale.
What is m
A. 35°
?
B. 325°
C. 145°
D. 45°
Circles
94. The center of the circle is S.
If m TRK equals 32.5°, what is m
A. 65°
B. 130°
C. 16.25°
D. 32.5°
TSK?
Y?
Similarity and Congruence
95. Kelly and Sandra each have a jewelry box in the shape of a rectangular prism, but they are different sizes. Kelly's
jewelry box has a volume of 720 cubic centimeters and is 6 centimeters tall. If Sandra's jewelry box has a base area of
480 square centimeters, what is the height of Sandra's jewelry box?
V = Bh
A. 24 inches
B. 36 inches
C. 12 inches
D. 48 inches
Similarity
96. Which pair of facts proves that
A.
T
C.
T
C and TU
B. TV
BA and
D. VT
BA and UV
C and
U
V
CB
TUV and
CBA are similar?
Note: picture not drawn to scale
A
CB
A
Line and Angle Relationships
97.
Which of the following are corresponding angles?
A. A and C
B.
C and
F
D.
A and
D
C.
A and
E
Congruence
98. Circle Q and circle W are congruent.
QR = 5.5 in
What is the area of circle W?
A. 21 in2
B. 30.25
C. 110.25
D. 11
in2
in2
in2
Distance, Midpoint, and Slope
99. What is the distance between the points (3, 7) and ((-1, 0) in the xy-plane?
A.
B.
C.
D.
Similarity
100. Which additional fact proves that
A. The measure of
Z is 58°.
B. The measure of
X is 58°.
D. The measure of
X is 68°.
C. The measure of
Z is 68°.
XYZ and RST are similar?
Note: picture not drawn to scale
Answers
1. C
2. B
3. B
4. C
5. B
6. C
7. B
8. B
9. B
10. B
11. D
12. A
13. A
14. B
15. A
16. C
17. D
18. C
19. D
20. C
21. B
22. C
23. C
24. C
25. B
26. A
27. C
28. B
29. D
30. C
31. C
32. C
33. C
34. D
35. A
36. B
37. A
38. C
39. A
40. B
41. B
42. B
43. C
44. B
45. A
46. A
47. B
48. C
49. B
50. B
51. A
52. B
53. A
54. D
55. B
56. B
57. C
58. B
59. B
60. C
61. C
62. C
63. C
64. D
65. C
66. B
67. A
68. B
69. C
70. D
71. C
72. D
73. C
74. D
75. A
76. C
77. A
78. C
79. C
80. B
81. B
82. C
83. D
84. A
85. B
86. B
87. B
88. A
89. A
90. A
91. C
92. B
93. B
94. A
95. C
96. C
97. C
98. B
99. C
100. B
Explanations
1. Given a right triangle and an acute angle of the right triangle, the following is true.
Thus, p can be found by the following steps.
2. To determine the length of the third side of an obtuse triangle, use the converse of the Pythagorean theorem, where c
is the longest side, as shown below.
If c2 < a2 + b2, the triangle is acute.
If c2 = a2 + b2, the triangle is right.
If c2 > a2 + b2, the triangle is obtuse.
Check each answer choice, with the two given sides, according to the rules above.
(21 cm)2 ? (15 cm)2 + (18 cm)2
441 cm2 ? 225 cm2 + 324 cm2
441 cm2 < 549 cm2
(24 cm)2 ? (15 cm)2 + (21 cm)2
576 cm2 ? 225 cm2 + 441 cm2
576 cm2 < 666 cm2
This is an acute triangle.
This is an acute triangle.
(21 cm)2 ? (15 cm)2 + (21 cm)2
441 cm2 ? 225 cm2 + 441 cm2
441 cm2 < 666 cm2
(27 cm)2 ? (15 cm)2 + (21 cm)2
729 cm2 ? 225 cm2 + 441 cm2
729 cm2 > 666 cm2
This is an acute triangle.
This is an obtuse triangle.
Therefore, 15 cm, 21 cm, and 27 cm are the side lengths of an obtuse triangle.
3.
Use the distance formula shown above and the coordinates of the points to determine the length of each side of the
figure.
The slopes of the sides can be determined by using the formula,
.
Since the figure has two sets of congruent, adjacent sides with reciprocal slopes, the most specific name for this shape
is a kite.
4. A cube net contains only squares.
A cylinder net contains two circles and one rectangle.
A sphere net is nearly impossible to make.
A cone net contains one circle and one partial larger circle.
Thus, a cone can be made with the net shown.
5. Since figure JKLM is an isosceles trapezoid, the diagonals are congruent. So, the length of diagonal JL is equal to the
length of diagonal KM, or 468 inches.
Since segment JP is perpendicular to side LM, triangle JLP is a right triangle, with right angle JPL. Use the
Pythagorean theorem to solve for the length of segment LP.
(JP)2 + (LP)2 = (JL)2
(180 inches)2 + (LP)2 = (468 inches)2
32,400 inches2 + (LP)2 = 219,024 inches2
(LP)2 = 186,624 inches2
LP = 432 inches
Since figure JKLM is an isosceles trapezoid, the non-parallel sides are congruent. So, the length of side JM is equal to
the length of side KL, or 204 inches.
Triangle JMP is a right triangle, with right angle JPM. Use the Pythagorean theorem to solve for the length of side PM.
(JP)2 + (PM)2 = (JM)2
(180 inches)2 + (PM)2 = (204 inches)2
32,400 inches2 + (PM)2 = 41,616 inches2
(PM)2 = 9,216 inches2
PM = 96 inches
Find the length of side LM.
LM = LP + PM
LM = 432 inches + 96 inches
LM = 528 inches
6. Use the distance formula to find the length of each side of the triangle.
Since two sides have the same length, the triangle is isosceles.
Where a, b, and c are the lengths of the sides and c is the longest side,
if c2 < a2 + b2, then it is an acute triangle,
if c2 = a2 + b2, then it is a right triangle, and
if c2 > a2 + b2, then it is an obtuse triangle.
Since the square of the longest side is equal to the sum of the squares of the other two sides, the triangle is right.
Therefore, the triangle is a right isosceles triangle.
7. Use the properties of 45°-45°-90° and 30°-60°-90° triangles to find the perimeter of the given figure.
Find the length of the hypotenuse of one 45°-45°-90° triangle.
length =
=
x
15
in
Find the length of the hypotenuse of one 30°-60°-90° triangle.
length = 2x
= 30 in
Find the length of the longer leg of one 30°-60°-90° triangle.
length =
Find the perimeter.
=
perimeter =
=
8.
15
(15
x
15
in
in + 15 in + 15
+ 30
in + 15
in + 30 in
+ 45) inches
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display ed. The file may hav e been
mov ed, renamed, or deleted. Verify
that the link points to the correct
file and location.
9. The formula to find the distance between two points in a coordinate plane is shown below.
Use this formula to solve the question.
10. Alternate exterior angles are congruent angles formed by two parallel lines cut by a transversal; located on opposite
sides of the transversal outside the parallel lines.
Lines 1 and 2 are parallel and they are intersected by line 3. Therefore,
B and G are alternate exterior angles.
11. The arc length of
can be found by using the formula below.
Arc length of
= m
A and
H are alternate exterior angles and
2
r
Arc length of
18
in
Arc length of
18
in
Arc length of
Therefore, the arc length of
equals
360
=
24
360
1
=
=
15
18
15
in
in.
12.
13. The formula for the surface area of the given rectangular prism is
SA = 2HW + 2HL + 2LW.
The prism has a height of H = 4 in, a length of L = 12 in, and a width of W = 3 in. Substitute the values into the surface
area formula.
SA =
=
=
=
2HW + 2HL + 2LW
2(4 in)(3 in) + 2(4 in)(12 in) + 2(12 in)(3 in)
24 in2 + 96 in2 + 72 in2
192 in2
14. Two lines that are parallel have the same slope.
Line r has a slope of -11. Therefore, line s has a slope of -11.
15. Make or think of the mat plan for the object.
Add the column sums to find the total number of cubes needed.
10 + 10 + 8 + 3 + 1 = 32
Therefore, 32 is the fewest number of cubes needed to build the object above.
16. In this picture,
O and
N are vertical angles.
Vertical angles are congruent; therefore, the measure of angle O is equal to the measure of angle N.
So, the measure of angle N is 45°.
17. The nets portrayed by X and Y will both create pentagonal pyramids when folded.
The net portrayed by W will create a truncated pentagonal pyramid.
The net portrayed by Z can be folded to create a pentagonal prism.
18. A 30°-60°-90° triangle is a special case of a right triangle.
In this problem, the height of the clock will equal the length of the longest leg of a 30°-60°-90° triangle. Start by
finding the value of .
Find the height of the clock.
So, the approximate height of the clock is
mm.
19. To determine the length of the third side of an acute triangle, use the converse of the Pythagorean theorem, where c
is the longest side, as shown below.
If c2 < a2 + b2, the triangle is acute.
If c2 = a2 + b2, the triangle is right.
If c2 > a2 + b2, the triangle is obtuse.
Check each answer choice, with the two given sides, according to the rules above.
(22 cm)2 ? (12 cm)2 + (18 cm)2
484 cm2 ? 144 cm2 + 324 cm2
484 cm2 > 468 cm2
(30 cm)2 ? (18 cm)2 + (22 cm)2
900 cm2 ? 324 cm2 + 484 cm2
900 cm2 > 808 cm2
This is an obtuse triangle.
This is an obtuse triangle.
(22 cm)2 ? (10 cm)2 + (18 cm)2
484 cm2 ? 100 cm2 + 324 cm2
484 cm2 > 424 cm2
(28 cm)2 ? (18 cm)2 + (22 cm)2
784 cm2 ? 324 cm2 + 484 cm2
784 cm2 < 808 cm2
This is an obtuse triangle.
This is an acute triangle.
Therefore, 18 cm, 22 cm, and 28 cm are the side lengths of an acute triangle.
20. If the base of a prism has n sides, use the expression 2n to find the number of vertices the prism has.
2(6) = 12
Therefore, a prism with a base that has 6 sides has 12 vertices.
faces
edges
vertices
6
12
8
n+2
3n
5
Triangular prism
Rectangular prism
9
7
Pentagonal prism
Base with n sides
15
6
10
2n
21. Find the slope of each side of the quadrilateral.
Since opposite sides have equal slopes, the opposite sides are parallel.
Since adjacent sides do not have negative reciprocal slopes, the sides are not perpendicular. So, the shape is not a
rectangle or a square.
Find the length of two adjacent sides.
Since the two adjacent sides are not equal, the shape is not a rhombus.
Therefore, the quadrilateral is a parallelogram.
22. Similar solids have the same shape but not necessarily the same dimensions. If two solids are similar, then the ratio
of the dimensions is a:b, or a/b.
Let x represent the front base edge of the bottom pyramid. Set up a ratio to determine the length of the side.
height of top pyramid
base edge of top pyramid
20 cm
28 cm
height of bottom pyramid
=
=
base edge of bottom pyramid
35 cm
x
20 cm · x = 28 cm · 35 cm
20 cm · x = 980 cm2
x = 49 cm
Therefore, the front base edge of the bottom pyramid is 49 cm.
23. When looking at this object from the front, you can determine that there are 5 columns. The column on the far left
is 1/2 cube tall. The next column over is 1 cube tall. The next one over is 2 cubes tall. The next one over is 2 cubes tall,
but the top cube will have a bisector across it due to the half cube in the middle row. The column on the right is 1 cube
tall.
24. To determine the side lengths of a right triangle, use the converse of the Pythagorean theorem, where c is the
longest side, as shown below.
If c2 < a2 + b2, the triangle is acute.
If c2 = a2 + b2, the triangle is right.
If c2 > a2 + b2, the triangle is obtuse.
Check each answer choice, according to the rules above.
(20 in)2 ? (8 in)2 + (16 in)2
400 in2 ? 64 in2 + 256 in2
400 in2 > 320 in2
(24 in)2 ? (16 in)2 + (20 in)2
576 in2 ? 256 in2 + 400 in2
576 in2 < 656 in2
This is an obtuse triangle.
This is an acute triangle.
(24 in)2 ? (12 in)2 + (16 in)2
576 in2 ? 144 in2 + 256 in2
576 in2 > 400 in2
(20 in)2 ? (12 in)2 + (16 in)2
400 in2 ? 144 in2 + 256 in2
400 in2 = 400 in2
This is an obtuse triangle.
This is a right triangle.
Therefore, 12 in, 16 in, and 20 in are the side lengths of a right triangle.
25. When point P(x,y) is reflected over the x-axis, the reflection point, P', is located at (x,-y), where only the sign of the
y-coordinate changes.
Thus, the coordinates of the reflection across the x-axis of point R are R'(7,2).
26. In this picture,
X and
O are alternate interior angles.
Alternate interior angles are congruent; therefore, the measure of angle X is equal to the measure of angle O.
So, the measure of angle O is 69°.
27. A regular polygon is a polygon that is equilateral and equiangular.
Polygon W is a triangle. A regular triangle always has three 60° angles. Since this triangle is a right triangle, it is not a
regular polygon.
Polygon Y is a quadrilateral. A regular quadrilateral is a square, which always has four 90° angles. Since this
quadrilateral has no right angles, it is not a regular polygon.
Polygon Z is a pentagon. A regular pentagon always has five 108° angles. Since this quadrilateral has two right angles,
it is not a regular polygon.
Polygon X is a hexagon. A regular hexagon always has six 120° angles and six congruent sides. Since this hexagon
matches this description, it is a regular polygon.
28. A 30°-60°-90° triangle is a special case of a right triangle.
In a 30°-60°-90° triangle, if the shorter leg is of length , then the longer leg is of length
In this problem, the shorter leg of the triangle measures
Therefore, the longer leg measures
.
inches.
inches.
29. A rectangular pyramid has 8 edges.
Triangular pyramid
Rectangular pyramid
Pentagonal pyramid
Base with n sides
faces
edges
vertices
5
8
5
4
6
n+1
6
10
2n
4
6
n+1
30. A 45°-45°-90° is a special case of a right triangle.
In a 45°-45°-90° triangle, if one leg of the triangle is of length , then the hypotenuse is of length
In this problem, one leg of the triangle measures 7 inches.
Therefore, the hypotenuse measures
inches.
31. The surface area of a globe (or sphere) is given by:
SA = 4 r2.
.
Since the diameter of each globe is 18 inches, the radius of each globe is 18 in ÷ 2 = 9 inches.
Substitute the radius into the formula above.
SA = 4 (9 in)2
= 1,017.36 in2
Since there are 5 globes in his classroom, the total surface area of all the globes is 5 × 1,017.36 in 2 = 5,086.8 in2.
32. A polygon with five vertices is called a pentagon.
33. If two triangles are congruent, their corresponding sides have equal lengths.
Since XZ corresponds to AC, the length of XZ is 58 units.
34. A 45°-45°-90° is a special case of a right triangle.
In a 45°-45°-90° triangle, if the hypotenuse of the triangle is of length
, then each leg is of length
.
In this problem, the hypotenuse of the triangle measures 6 inches.
Therefore, each leg measures
35. Notice that
FDG and
Also, notice that
FDG and
inches.
FEG are inscribed angles.
FEG intercept the same arc, FG.
If two inscribed angles intercept the same arc, then they have the same measure.
So, m
FDG = m
FEG = 30°.
36. Since figure QRST is congruent to figure WXYZ,
Q is congruent to
W.
Find the m Q.
Adjacent angles of a parallelogram are supplementary.
Since m R = 107°, m Q = 180° - 107° = 73°.
Therefore, m W is 73°.
37. Congruent solids have the same shape and the same dimensions.
Use the area formula to determine the radius of the base of cone F, which is also the radius of the base of cone E.
A = r2
625 cm2 = r2
625 cm2 = r2
25 cm = r
To find the slant height l, use the Pythagorean theorem. The radius and height of the cone are the legs of the right
triangle and the slant height is the hypotenuse.
r 2 + h2 = l 2
(25 cm)2 + (60 cm)2 = l2
625 cm2 + 3,600 cm2 = l2
4,225 cm2 = l2
65 cm = l
Now, substitute the radius and slant height into the surface area formula.
SA = rl + r2
= (25 cm)(65 cm) + (25 cm)2
= 1,625 cm2 + 625 cm2
= 2,250 cm2
Therefore, the surface area of cones E and F is 2,250 cm2.
38. Two solids are congruent if corresponding angles are congruent, corresponding edges are congruent, corresponding
faces are congruent, and their volumes are equal.
Since the volume of the box of cookies is 285 in 3, the volume of the box of crackers is also 285 in 3.
So, set up an equation using the area of the square base and the volume, and solve for the length, x.
V
285 in3
285 in3
285 in3
11.4 in
=
=
=
=
=
(area of base)(height)
(y2)(z)
(w2)(x)
(25 in2)(x)
x
Therefore, the length of the box of crackers is 11.4 in.
39. In the right triangle given in the problem, the two legs have the same length. This indicates that the triangle is a 45°45°-90° triangle, or isosceles right triangle.
Since
MNO is one of the congruent base angles, m
MNO is 45°.
40. Each of the pencils that made up the sides of the original triangle were broken in half to form the new triangle. The
sides of the new triangle are half the length of the sides of the original triangle, so the corresponding side lengths are
proportional.
Therefore, the triangles are similar by SSS.
41. Use the Pythagorean theorem.
42. Since PQRS is a kite, PQ
RQ and PS
RS.
QS SQ by the reflexive property. Therefore, PQS
congruent triangles are congruent. So, PT = RT = 8 cm.
RQS by SSS. Thus, PT
RT since altitudes of
Since QS = 21 cm and the ratio QT to TS is 2 to 5,
QT = 6 cm and TS = 15 cm.
By the Pythagorean Theorem,
PQ2 = PT2 + QT2
PQ2 = (8 cm)2 + (6 cm)2
PQ2 = 64 cm2 + 36 cm2
PQ2 = 100 cm2
PQ = 10 cm
and
PS2 = PT2 + TS2
PS2 = (8 cm)2 + (15 cm)2
PS2 = 64 cm2 + 225 cm2
PS2 = 289 cm2
PS = 17 cm
Since similar parts of congruent triangles are congruent,
RQ = PQ = 10 cm and RS = PS = 17 cm.
Therefore,
Perimeter of PRQS = PQ + RQ + PS + RS
= 10 cm + 10 cm + 17 cm + 17 cm
= 54 cm
43. Congruent solids have the same shape and the same dimensions.
Use the volume formula to determine the height of rectangular prism B, which is also the height of rectangular prism A.
V = lwh
324 cm3 = (9 cm)(3 cm)h
324 cm3 = (27 cm2)h
12 cm = h
Therefore, the height of rectangular prisms A and B is 12 cm.
44. Given that figure ABCD is a rhombus, diagonal AC and diagonal BD must bisect opposite angles.
Therefore, m DAE = m BAE.
Given that m DAE = m BAE and m DAB = 132°, the following is true.
m DAE + m BAE = m DAB
m DAE + m BAE = 132°
m BAE + m BAE = 132°
2 × m BAE = 132°
m BAE = 66°
Since segment AC and segment BD are diagonals, it is also true that segment AC is perpendicular to segment BD;
therefore, m AEB = 90°.
To find the measure of
ABE, use the triangle ABE.
m BAE + m AEB + m ABE = 180°
66° + 90° + m ABE = 180°
156° + m ABE = 180°
m ABE = 24°
The measure of
ABE equals 24°.
45. Since the sum of the interior angles of a triangle equals 180°, solve the following for x.
m R + m S + m T = 180°
2x - 7° + 4x - 6x + 40° + 18x + 3° = 180°
6x2 + 12x + 36° = 180°
x2 + 2x + 6° = 30°
x2 + 2x - 24° = 0°
(x + 6°)(x - 4°) = 0°
x = (-6)°, 4°
2
2
Since all of the angles are positive angles, x = 4°.
The m Q is found by substituting x = 4° into the equation representing m R and subtracting from 180°.
180° - m R = 180° - [2(x)2 - 7°]
= 180° - [2(4°)2 - 7°]
= 180° - 2(16°) + 7°
= 155°
46. The ASA theorem states that if two angles and the included side of one triangle are congruent to two angles and the
included side of another triangle, then the two triangles are congruent.
Lisa's drawing does not prove congruence by the ASA theorem because the congruent sides are not included
between the congruent angles.
47. Since line TQ is tangent to circle S at point Q and chord RQ is a diameter of circle S, line TQ is perpendicular to
chord RQ. Thus, RQT is a right angle.
So,
RQP and
PQT are complementary angles.
m
RQP + m
27° + m
PQT = 90°
PQT = 90°
m
m
PQT = 90° - 27°
PQT = 63°
48. Corresponding parts of congruent triangles are congruent. The following congruencies are true for this set of
triangles.
LM
US
LN
UT
MN
Therefore, the true statement is LN
ST
UT.
49. Let x represent the length of one side of the Plumbing building, y represent the length of one side of the A/C
building, and z represent the length of one side of the Electrical building.
Then, x2 is the area of the Plumbing building, y2 is the area of the A/C building, and z2 is the area of the Electrical
building.
Now, use the Pythagorean theorem.
= z2
3,600 ft2 + 625 ft2 = z2
4,225 ft2 = z2
x2 + y2
So, the area of the Electrical building is 4,225 ft2.
50. The length of the diagonal of the illuminated area will be the hypotenuse of the triangle formed.
Since the angle at the ground and the diagonal are known, the cosine equation should be used.
51.
The link ed image cannot be
display ed. The file may hav e been
mov ed, renamed, or deleted. Verify
that the link points to the correct
file and location.
52. The AAS corollary states that if two angles and a non-included side of one triangle are congruent to two angles and
a non-included side of another triangle, then the two triangles are congruent.
In the drawing, Fred marked the included side on each triangle as congruent; therefore, the triangles are not
congruent by the AAS corollary.
53. Two lines that are parallel have the same slope.
Line p has an undefined slope.
Therefore, line q has an undefined slope.
54. In the right triangle, the tangent of
A is the ratio of the side opposite
A to the side adjacent
A.
55. Given a right triangle and an acute angle
of the right triangle, the following is true.
Thus, z can be found by the following steps.
56. Let x represent the distance traveled by Car J, y represent the distance traveled by Car K, and z represent the
distance between the two cars. Now, use the Pythagorean theorem.
x2 + y2 = z2
(84 miles)2 + y2 = (105 miles)2
7,056 miles2 + y2 = 11,025 miles2
y2 = 11,025 miles2 - 7,056 miles2
y2 = 3,969 miles2
y = 63 miles
So, Car K traveled 63 miles west.
57. Use the distance formula to find the length of each side of the triangle.
Since two sides have the same length, the triangle is isosceles.
Where a, b, and c are the lengths of the sides and c is the longest side,
if c2 < a2 + b2, then it is an acute triangle,
if c2 = a2 + b2, then it is a right triangle, and
if c2 > a2 + b2, then it is an obtuse triangle.
Since the square of the longest side is less than the sum of the squares of the other two sides, the triangle is acute.
Therefore, the triangle is an acute isosceles triangle.
58. If you were looking at this object from the right side, you would see that the back row is 3 cubes high (at its tallest
point), and the front row is 2 cubes high (at its tallest point).
If you were below this object looking up at it, you would only see the faces of the cubes that are on the bottom. You
would see 4 cubes across in the front and 3 cubes across in the back.
59. Similar solids have the same shape but not necessarily the same dimensions. If two solids are similar, then the ratio
of the dimensions is a:b, or a/b.
Set up a ratio to determine the width of the larger base. Let x represent the larger base length.
small base length
small base width
9 cm
14 cm
=
=
large base length
large base width
18 cm
x
9 cm · x = 14 cm · 18 cm
9 cm · x = 252 cm2
x = 28 cm
Use the formula for area and the values of the length and width of the larger prism.
Area = 18 cm · 28 cm
Area = 504 cm2
Therefore, the area of the larger base is 504 cm2.
60. The formula for the area of a trapezoid is shown below.
Let side AB be base1. The length of side AB is given.
Let side EC be base2. Calculate the length of side EC.
The height of the trapezoid is given.
Find the area of the trapezoid.
61. In a right triangle, the sine of an acute angle is equal to the length of the opposite side divided by the length of the
hypotenuse.
Use the information given in the problem to find sin(45°).
62. First convert degrees to radians.
Now use the formula for arc length.
= 180°
×
radians
180°
=
radians
s=
r
= (6 inches)( )
= 6 inches
63. Vertical angles are congruent opposite angles formed by two intersecting lines.
Lines 1 and 2 are parallel and they are intersected by line 3. This gives four sets of vertical angles.
A and D
B and C
E and H
F and G
64. From the front, the left column must be 3 cubes high, the middle two columns must be 1 cube high, and the right
column must be 2 cubes high. Y, W, and Z fit the profile. Thus, X is eliminated.
From the right side, the object will be 2 cubes tall toward the front, 1 cube tall in the middle, and 3 cubes tall towards
the back. Both Y and W fit the side profile. Thus, Z is eliminated.
From the top view, we see that the left column has 3 cubes, which both Y and W have. The 2nd to the left column has 2
cubes with a space in the middle, which both Y and W have. The 2nd to the right column has 2 cubes, which only Y
has.
Therefore, only Y fits the description.
65. Count the number of faces for the hexagonal prism.
There are 2 hexagonal bases and 6 rectangular lateral faces. So, a hexagonal prism has 8 faces.
Count the number of edges for the hexagonal prism.
There are 6 edges for each base and 6 edges for the lateral faces. So, a hexagonal prism has 18 edges.
Count the number of vertices for the hexagonal prism.
There are 6 vertices for each base. So, a hexagonal prism has 12 vertices.
Sum = faces + edges + vertices
= 8 + 18 + 12
= 38
66. A rectangular prism has six rectangular faces.
A rectangular pyramid has a rectangular base and four triangular faces.
A triangular pyramid has four triangular faces.
A triangular prism has two triangular bases and three rectangular faces.
So, the net pictured will form a triangular prism.
67. The bottom of the object has 6 cubes. The middle has 3 cubes. The top has 1 cube.
68. Since ML
MN and LP
NP, the ratios created from the congruence statements are proportional.
69. The sum of the angles of a triangle is 180°. Both triangles are right triangles.
In the larger triangle, the measure of
m is 49°, so the measure of
Similarly, in the smaller triangle, the measure of
n must be 180 - 90 - 49 = 41°.
q is 41°, so the measure of
p must be 180 - 90 - 41 = 49°.
Since the triangles have congruent corresponding angles, they are similar by AA.
70. Alternate interior angles are congruent angles formed by two parallel lines cut by a transversal; located on opposite
sides of the transversal inside the parallel lines.
Lines 1 and 2 are parallel and they are intersected by line 3. Therefore,
D and E are alternate interior angles.
71. In the right triangle, the cosine of
C and
B is the ratio of the side adjacent
F are alternate interior angles and
B to the hypotenuse.
72. The Angle-Angle-Side rule states that if two angles and a non-included side of a triangle are congruent to two angles
and the corresponding non-included side of another triangle, then the two triangles are congruent.
It is given that H
K. Since the two triangles form a pair of vertical angles at point J, these triangles have two
adjacent angles that can be used to prove congruency. To use the AAS rule, one of the pairs of sides that is not
inclusive in the adjacent angles must be proven congruent.
Sides GH
LK or sides GJ
LJ could be used to prove congruency by the AAS rule.
So, the answer is III or IV only.
73. If the base of a prism has n sides, use the expression 3n to find the number of edges the prism has.
3(6) = 18
Therefore, a prism with a base that has 6 sides has 18 edges.
Triangular prism
faces
5
edges
9
vertices
6
Rectangular prism
6
12
Base with n sides
n+2
3n
Pentagonal prism
7
15
8
10
2n
74. Let x represent how far away the foot of the ladder should be from the building, y represent how far up the building
the ladder needs to reach, and z represent the length of the ladder.
Now, use the Pythagorean theorem.
= z2
x2 + (36 ft)2 = (39 ft)2
x2 + 1,296 ft2 = 1,521 ft2
x2 = 1,521 ft2 - 1,296 ft2
x2 = 225 ft2
x = 15 ft
x2 + y2
So, the foot of the ladder needs to be placed 15 feet away from the building.
75. If two triangles are similar, then the ratios between the lengths of corresponding sides are equivalent. If triangles
JLK and PRQ are similar, then the following ratios are equivalent.
Therefore, the equation which is true is
.
76. Since the coolers are congruent, their dimensions and surface areas are congruent.
In this problem, the value of B is the area of a face of either cooler.
Use the given value of the surface area of Michelle's cooler to solve for the side length of either cooler.
SA = 6B
486 square inches = 6(side)(side)
81 square inches = (side)2
9 inches = side
77. A 45°-45°-90° triangle, also known as an isosceles right triangle, is a special case of a right triangle.
In a 45°-45°-90° triangle, if a leg of the triangle has length
, then the hypotenuse has length
In this problem, the side length of the baseball field measures 90 ft.
.
Therefore, the distance from third base to first base is
78. In this picture,
N and
ft.
Y are alternate exterior angles.
Alternate exterior angles are congruent; therefore, the measure of angle N is equal to the measure of angle Y.
So, the measure of angle Y is 54°.
79. Find the slope of each side of the quadrilateral.
Since opposite sides have equal slopes, the opposite sides are parallel. So, the shape is not a trapezoid.
Since adjacent sides have slopes that are negative reciprocals, the sides are perpendicular. So, the shape is either a
rectangle or a square.
Find the length of two adjacent sides.
Since the two adjacent sides are not equal, the shape is not a square.
Therefore, the quadrilateral is a rectangle.
80. Since the diameter of the sphere is 18 cm, the radius of the sphere is 9 cm.
Substitute the radius into the formula for surface area of a sphere.
4
r2
=
=
4
(9 cm)2
4
(81 cm2)
1,017.88 cm2
81. When point P(x,y) is translated 6 units to the right and then translated 9 units down, the translation point, P', is
located at (x + 6,y - 9), where 6 units are added to the x-coordinate to move right and 9 units are subtracted from the ycoordinate to move down.
Thus, the coordinates of point S translated 6 units to the right and then translated 9 units down are S(5,-6).
82. Use the Pythagorean theorem.
83. Angles RQS and SQT are supplementary angles.
180° - 70° = 105°
Thus m
RQS = 105°.
The measure of a minor arc is equal to the measure of its related central angle.
m
=m
RQS = 105°
To find the measure of a major arc, subtract the measure of the minor arc from 360°.
m
Therefore, m
= 360° - 105° = 255°
equals 255°.
84. If point E is the midpoint of AD and point B is the midpoint of AC, then AE is half the length of AD and AB is half
the length of AC. The picture also shows that both triangles share angle A.
Since the measures of two sides of one triangle are proportional to the measures of two corresponding sides of the other
triangle and the included angles are congruent, the triangles are similar by SAS.
85. Use the formula for volume of a rectangular prism.
Therefore, the height of the rectangular prism is 6 in.
86. A secant line is a line which passes through at least two points of a curve. Examine the picture and note that there
are two secant lines which pass through this circle.
The angle created by two secant lines has a measure that is equal to half of the difference between the corresponding
arc angles. For this circle, this comes to
m
BCD = 1/2(m
-m
)
So, use the given arc measurement and the angle measurement to solve for m
m
BCD
=1
29° = 1
58° =
105° =
/2(m
-m
/2(m
- 47°)
m
.
)
- 47°
m
87. Since the logo is a right triangle, use the Pythagorean Theorem to find the third side.
a2 + b2 = c2
92 + 122 = c2
225 = c2
15 = c
The perimeter of the small logo is 9 cm + 12 cm + 15 cm = 36 cm.
Since the logo will be painted on the bike shop wall 15 times bigger, multiply the perimeter of the small logo by 15.
15 × 36 cm = 540 cm
So, Brandon will need a bike chain that is 540 cm long to outline the logo painted on the bike shop wall.
88. Let
be the angle of elevation in the right triangle. Therefore, the following is true.
The question gives that the hypotenuse represents the distance between Daniel and the top of the lighthouse. Label this
variable d.
It also gives that the side opposite
is equal to 140 feet and
= 50°.
Therefore, the following is true.
Now, solve for d.
Daniel is about 183 feet from the top of the lighthouse.
89. Since the circles are congruent, their circumferences and areas are equal.
Use the formula for area to solve for the value of r.
A=
r2
50.24 sq in = 3.14r2
16 sq in = r2
4 in = r
Now, use the value of r in the formula for circumference.
C=
=
2
r
2(3.14)(4 in)
25.1 inches
90. For any given angle
, the new coordinates for a rotation about the origin of
are the following.
(x' , y') = (x cos
)
- y sin
, x sin
+ y cos
A positive angle of rotation turns the figure counterclockwise, and a negative angle of rotation turns the figure in a
clockwise direction.
The polygon has been transformed by a clockwise rotation of 270° about the origin.
(x' , y') = (x cos(-270°) - y sin(-270°) , x sin(-270°) + y cos(-270°))
= (x(0) - y(1) , x(1) + y(0))
= (-y , x)
91. A 30°-60°-90° triangle is a special case of a right triangle.
In a 30°-60°-90° triangle, if the shorter leg is of length a, then the hypotenuse is of length 2a.
In this problem, the shorter leg of the triangle measures
Therefore, the hypotenuse measures
92. In this picture,
O and
inches.
inches.
Y are corresponding angles.
Corresponding angles are congruent; therefore, the measure of angle O is equal to the measure of angle Y.
So, the measure of angle Y is 65°.
93. The measure of a minor arc is equal to the measure of its related central angle.
m
=m
SQP = 35°
To find the measure of a major arc, subtract the measure of the minor arc from 360°.
m
Therefore, m
= 360° - 35° = 325°
equals 325°.
94. The inscribed angle of a circle is equal to one-half the measure of the arc it intercepts.
Multiply the measure of the inscribed angle by two to find the measure of the central angle.
m
TRK = 32.5°
2(32.5°) = 65°
Therefore, m
TSK equals 65°.
95. Similar solids have the same shape but not necessarily the same dimensions. If two solids are similar, then the ratio
of their dimensions is a:b, or a/b. Their areas have a ratio of a2:b2 or a2/b2.
Use the given volume formula to determine the area of the base of Kelly's jewelry box.
Since Kelly's jewelry box has a base area of 120 cm2 and Sandra's jewelry box has a base area of 480 cm 2, the ratio of
their base areas is 120 cm2:480 cm2, which reduces to 1:4. So, the ratio of their heights (dimensions) is
, or 1:2.
Set up a ratio to determine the height of Sandra's jewelry box, h.
So, the height of Sandra's jewelry box is 12 cm.
96. If the two triangles are similar, then corresponding angles are congruent. Similar triangles do not necessarily have
congruent sides.
Therefore, the following facts prove that
T
C
U
B
V
A
TUV and
T
CBA are similarby the AA Similarity Postulate.
C and
V
A
97. Corresponding angles are congruent angles formed by two parallel lines cut by a transversal; positioned on
matching "corners" of intersections.
Lines 1 and 2 are parallel and they are intersected by line 3 giving four sets of corresponding angles.
A and
B and
C and
D and
E
F
G
H
98. Since the circles are congruent, their dimensions and areas are equal.
Find the area of circle Q.
area =
r2
=
=
(5.5 in)2
30.25
in2
area of circle Q = area of circle W
Therefore, the area of circle W is 30.25
in2.
99. If the given ordered pairs are graphed on a coordinate plane, they will form the hypotenuse of a right triangle. Use
the lengths of the legs of the right triangle to find the length of the hypotenuse.
Subtract the x-coordinates to find the length of one leg.
3 - (-1) = 4
Subtract the y-coordinates to find the length of the other leg.
7-0=7
Now, the Pythagorean theorem can be used to find the length of the hypotenuse, which is the distance between the two
points.
a2 + b2 = c2
42 + 72 = c2
16 + 49 = c2
65 = c2
=
Therefore, the distance between the two points is
100. To determine which fact proves that
c
units.
XYZ and
RST are similar, first determine the measure of
Use the fact that the sum of the angles is a triangle is 180°.
m
R+m
m
S+m
T
R + 68° + 54°
= 180°
= 180°
= 58°
m
R
If the two triangles are similar, then corresponding angles will be congruent, and will have the same measure.
R.
X
R
m
X = 58°
Y
S
m
Y = 68°</
Z
T
m
Z = 54°</
Therefore, the following fact proves that the two triangles are similar by the AA Similarity Postulate.
the measure of
X is 58°