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# Geometry Review Practice Test

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

## Geometry Review Practice Test

You may draw diagrams or use a calculator if necessary. A good score is at least 75 correct. It’s best to have a friend check your score the first time, so you won’t memorize the answers if you want to take the test again.

1. Imagine two triangles such that their corresponding sides have equal lengths as you proceed around them both in the same direction. These two triangles are

(a) isosceles

(b) non-Euclidean

(c) equilateral

(d) directly congruent

(e) symmetrical

2. Suppose a rhombus has diagonals of equal length. From this we can conclude that the rhombus is

(a) a rectangle

(b) irregular

(c) a square

(d) a trapezoid

(e) none of the above

3. Look at Fig. Exam-1. Which, if any, of the following statements (a), (b), (c), or (d) is true?

(a) Δ PSR is an equilateral triangle

(b) Line segments PS and SR are equally long

(c) Δ PSR is congruent to Δ PSQ

(d) QPS = QSP

(e) None of the above statements is true

Fig. Exam-1 . Illustration for Questions 3, 4, and 5 in the final exam.

4. In Fig. Exam-1, the measure of RQS is

(a) equal to 2 π rad minus the sum of the measures of PSQ and RSQ

(b) equal to 180° minus the measure of PQS

(c) equal to twice the measure of PSQ

(d) equal to half the sum of the measures of PSQ and RSQ

(e) not equal to anything described above

5. Fill in the blank: In Fig. Exam-1, line QS is ___ line segment PR .

(a) parallel to

(b) congruent to

(c) a perpendicular bisector of

(d) a parallel bisector of

(e) in a different plane than

6. The formula for the interior area, A (in square units) of a regular polygon with n sides of length s is:

A = ( ns 2 /4) cot (180°/ n )

What is the interior area of a regular heptagon (7-sided polygon) with sides each measuring 1.000 meter in length? You may use a calculator to determine this. Express the answer to two decimal places. The cotangent (cot) of an angle is equal to the cosine (cos) divided by the sine (sin).

(a) 0.84 square meters

(b) 3.63 square meters

(c) 5.32 square meters

(d) 8.33 square meters

(e) 10.00 square meters

7. Refer to Fig. Exam-2. If r 1 = r 2 , then

(a) the interior area of the figure is equal to πr 2 2

(b) the interior area of the figure is equal to 2 r 2 2

(c) the perimeter of the figure is equal to 2 r 1

(d) the perimeter of the figure is equal to πr 2

(e) none of the above

8. In Fig. Exam-2, the ratio of r 1 , the length of the major semi-axis, to r 2 , the length of the minor semi-axis, is called the

(a) elongation

(b) eccentricity

(c) ellipticity

(d) deviation

(e) oblongation

Fig. Exam-2 . Illustration for Questions 7, 8, and 9 in the final exam.

9. Refer to Fig. Exam-2. The interior area, A , of the ellipse is given by:

A = πr 1 r 2

Based on this formula, if the length of the major semi-axis of an ellipse is tripled, what must happen to the length of the minor semi-axis in order for the interior area to remain the same?

(a) We cannot say, because the formula does not contain enough information

(b) The minor semi-axis must become 1/3 as great

(c) The minor semi-axis must become 1/6 as great

(d) The minor semi-axis must become 1/9 as great

(e) The minor semi-axis must become 1/27 as great

10. In Euclidean plane geometry, how many points are required to uniquely define a single straight line?

(a) None

(b) One

(c) Two

(d) Three

(e) Four

11. Imagine that you have a telescope equipped with a camera. You focus on a distant, triangular sign and take a photograph of it. Then you triple the magnification of the telescope and, making sure the whole sign fits into the field of view of the camera, you take another photograph. When you get the photos developed, you see triangles in each photograph. No matter what else might be true about this scenario, we can conclude for certain that the two triangles in the photographs must be

(a) equilateral

(b) symmetrical

(c) non-Euclidean

(d) isosceles

(e) none of the above

12. A rhombus is a geometric figure in which

(a) all the sides are equally long

(b) all the angles have the same measure

(c) the sum of the measures of the interior angles is 720°

(d) no two points lie in the same plane

(e) at least one of the sides is infinitely long

13. Refer to Fig. Exam-3. What is the equation of line L ?

(a) − x + y − 5 = 0

(b) − x + y + 5 = 0

(c) x + y + 5 = 0

(d) −2 x + 3 y − 1 = 0

(e) 2 x − 3 y + 1 = 0

Fig. Exam-3 . Illustration for Questions 13, 14, and 15 in the final exam.

14. Refer to Fig. Exam-3. What is the equation of line M ?

(a) 3 xy = 0

(b) x − 3 y + 6 = 0

(c) x + 3 y + 6 = 0

(d) − x + 3 y + 6 = 0

(e) −3 x − 3 y = 0

15. Refer to Fig. Exam-3. Imagine curves L and M as infinitely long, straight lines. Do these lines intersect? If so, what are the coordinates ( x 0 , y 0 ) of their intersection point?

(a) The lines intersect at ( x 0 , y 0 ) = (−21,−11)

(b) The lines intersect at ( x 0 , y 0 ) = (−21/2,−11/2)

(c) The lines intersect at ( x 0 , y 0 ) = (−10,−5)

(d) The lines intersect, but more information is needed to figure out where

(e) The lines do not intersect

16. What is the distance r between the points (−3,−3) and (−6,−7) on the Cartesian plane?

(a) r = 3

(b) r = 4

(c) r = 5

(d) r = (−9,−10)

(e) r = (3,4)

Fig. Exam-4 . Illustration for Questions 17 and 18 in the final exam.

17. Examine Fig. Exam-4. Suppose planes Y and Z are parallel, and they intersect plane X in lines L and M , respectively. Suppose line PQ is in plane X , and is perpendicular to both lines L and M . Suppose line RS is in plane Y , and is perpendicular to line L . Suppose line TU is in plane Z , and is perpendicular to line M . Let V and W be intersection points among the lines, as shown. From these facts, we can conclude that

(a) PWT = PVS

(b) PWT = QVR

(c) PWT = QVS

(d) PWT = QWT

(e) PWT = UWP

18. In Fig. Exam-4, assume all of the conditions described in Question 17 hold true. Which of the following are corresponding angles?

(a) PWT and WVR

(b) PWT and WVS

(c) QWU and QWT

(d) QWU and QVR

(e) QVS and PVR

19. Vertical (that is, opposite) dihedral angles between two intersecting planes always have measures that

(a) add up to 90°

(b) add up to 180°

(c) add up to 360°

(d) are the same

(e) none of the above

20. A cube has

(a) 6 faces, 12 edges, and 12 vertices

(b) 6 faces, 8 edges, and 8 vertices

(c) 6 faces, 12 edges, and 8 vertices

(d) 8 faces, 8 edges, and 8 vertices

(e) 8 faces, 12 edges, and 12 vertices

21. In cylindrical coordinates, the position of a point is defined according to

(a) two angles and a distance

(b) two distances and an angle

(c) three distances

(d) three angles

(e) none of the above

22. The equation x 2 + y 2 = 1 can be used to define

(a) the exponential function

(b) the logarithmic function

(c) the cosine function

(d) the hyperbolic function

(e) none of the above

23. The tangent function y = tan x is defined for all the following values of x except:

(a) x = 0°

(b) x = 45°

(c) x = 90°

(d) x = 135°

(e) x = 180°

24. Imagine a vector a that has magnitude of 3 and points straight up (elevation 90°), and a vector b that has magnitude 4 and points toward the western horizon (azimuth 270°, elevation 0°). The cross product of a and b , written a × b , is

(a) a scalar equal to 12

(b) a scalar equal to 0

(c) a vector with magnitude 12, pointing toward the west but above the horizon

(d) a vector with magnitude 12, pointing toward the southern horizon

(e) impossible to determine without more information

25. Imagine a vector a that has magnitude of 3 and points straight up (elevation 90°), and a vector b that has magnitude 4 and points toward the western horizon (azimuth 270°, elevation 0°). The dot product of a and b , written a • b , is

(a) a scalar equal to 12

(b) a scalar equal to 0

(c) a vector with magnitude 12, pointing toward the west but above the horizon

(d) a vector with magnitude 12, pointing toward the southern horizon.

(e) impossible to determine without more information

26. When using a drafting compass and straight edge to perform a geometric construction, you must

(a) not make use of any calibrated scales

(b) always use a pen, not a pencil, to make markings on the paper

(c) use both instruments at least once

(d) never draw circles of arbitrary radius

(e) never draw line segments of arbitrary length

27. A circle is a specific type of

(a) polygon with infinitely many sides

(b) ellipse

(c) cone

(d) parabola

(e) none of the above

28. Refer to Fig. Exam-5. The top and the base of the figure are circles, with center points P and Q , as shown. This object is called a

(a) trapezoidal cone

(b) trapezoidal cylinder

(c) frustum of a cone

(d) partial cone

(e) truncated cylinder

29. Let S be the slant surface area of the object in Fig. Exam-5, that is, the area not including the base or the top. Let s be the slant height, r 1 be the radius of the circular top, and r 2 be the radius of the circular base. Let P be the center of the circular top, and Q be the center of the circular base. Let h be the height of the figure. The value of S is given by the following formula:

S = πs ( r 1 + r 2 )

where π is approximately equal to 3.14159. What happens to the slant surface area if all the dimensions of this object are doubled?

(a) It does not change

(b) It doubles

(c) It becomes four times as great

(d) It becomes eight times as great

(e) It is impossible to say without more information

Fig. Exam-5 . Illustration for Questions 28, 29, and 30 in the final exam.

30. Refer to Fig. Exam-5. Suppose that r 2 = 2 r 1 . Imagine some plane X that contains the line segment connecting points P and Q . The intersection between plane X and the surface of the object, including the base and the top, is

(a) a triangle

(b) a rhombus

(c) a rectangle

(d) a parallelogram

(e) none of the above

31. Suppose a triangle has a base length of 10 meters and a height of 6 meters. The interior area of this triangle is

(a) approximately 7.75 square meters

(b) 20 square meters

(c) 30 square meters

(d) 60 square meters

(e) impossible to determine without more information

32. Suppose a triangle has a base length of 10 meters, and the other two sides both measure 8 meters in length. How long is the line segment that joins the midpoints of the sides that are 8 meters long?

(a) 2 meters

(b) 4 meters

(c) 5 meters

(d) 8 meters

33. Suppose a triangle has a base length of 7 meters, and the other two sides both measure 11 meters in length. The line segment that joins the midpoints of the sides that are 11 meters long is

(a) perpendicular to those two sides

(b) parallel to those two sides

(c) normal to the base

(d) isosceles to the base

(e) parallel to the base

34. When two planes intersect, the measures of the adjacent dihedral angles defined by the intersection add up to

(a) 90°

(b) 180°

(c) 270°

(d) 360°

(e) 540°

35. Consider the mean radius of the earth to be 6371 kilometers. Consider the mean radius of the sun to be 696,000 kilometers. The volume of the sun is

(a) approximately 109 times the volume of the earth

(b) approximately 11,900 times the volume of the earth

(c) approximately 1,300,000 times the volume of the earth

(d) approximately 142,000,000 times the volume of the earth

(e) impossible to determine without more information

36. Imagine a torus T whose inner radius is half its outer radius. Suppose a line L passes through the center point of T . The intersection of L and T is

(a) four points

(b) three points

(c) two points

(d) the empty set (no points)

(e) impossible to determine without more information

37. A vector that begins at the origin of a coordinate system, and points outward in a specific direction (or orientation) from there, is said to be

(a) in unit form

(b) in zero form

(c) in real-number form

(d) in Euclidean form

(e) in standard form

38. Suppose we set off on a bearing of 90° in the navigator’s polar coordinate system. We stay on a straight course. If the starting point is considered the origin, what is the graph of our path in Cartesian coordinates?

(a) y = x , where x ≤ 0

(b) y = 0, where x ≥ 0

(c) x = 0, where y ≥ 0

(d) y = x , where x ≥ 0

(e) None of the above

39. Skew lines

(a) are parallel, but they intersect each other

(b) are orthogonal, but they do not intersect each other

(c) are not parallel, and they lie in different planes

(d) are not parallel, but they lie in a single plane

(e) none of the above

40. If two lines intersect and are perpendicular, then they

(a) lie in different planes

(b) are parallel

(c) are geodesics on a sphere

(d) lie in the same plane

(e) have a common center point

41. Consider a circle C that is inscribed by a regular polygon S having n sides, and that is circumscribed by a regular polygon T, also having n sides. As n increases without limit, what happens to the perimeters of S and T ?

(a) Their ratio approaches 1:1

(b) Their difference becomes greater and greater

(c) Their product approaches 1

(d) Their ratio approaches 1: π

(e) Their ratio approaches l:2 π

42. What is the slope of the line represented by the equation y − 2 = 4( x + 5)?

(a) −2

(b) 2

(c) 4

(d) 5

(e) 20

43. Right ascension is measured eastward from the position of the sun in the heavens on approximately

(a) March 21

(b) June 21

(c) September 21

(d) December 21

(e) the middle of the period of daylight

44. Right ascension, declination, and radius together comprise a scheme of

(a) polar coordinates

(b) Cartesian coordinates

(c) spherical coordinates

(d) cylindrical coordinates

(e) logarithmic coordinates

45. The 3D surface of a 4D sphere (or four-sphere) is an example of

(a) a 2D space

(b) a finite but unbounded 3D space

(c) an infinite but bounded 3D space

(d) a finite, bounded 3D space

(e) none of the above

46. What are the points, if any, at which the circle x 2 + ( y − 1) 2 = 1 intersects the y axis?

(a) It is impossible to tell without more information

(b) (0,0) and (0,2)

(c) (0,0) and (0,−2)

(d) (0,0)

(e) There are none

47. A triangle on a surface can have interior angles, each of whose measure is between

(b) 0 rad and π /2 rad

(d) 0 rad and 2 π rad

(e) π rad and 2 π rad

48. Each interior angle of a regular hexagon measures

(a) π /6 rad

(b) π /3 rad

(c) π /2 rad

(d) 2 π /3 rad

(e) 3 π /2 rad

49. The time equivalent of 1000 kilometers, using the speed of light (300,000,000 meters per second) as a standard, is approximately

(a) 333 second-equivalents

(b) 33.3 second-equivalents

(c) 3.33 second-equivalents

(d) 0.333 second-equivalents

(e) none of the above

50. Suppose there is a geodesic L on a surface S . Let P be some point near, but not on, the geodesic L . Suppose there exist infinitely many geodesics M 1 , M 2 , M 3 ,. . . on the surface S that pass through point P and do not intersect geodesic L . From this we know that

(a) the surface S is non-Euclidean

(b) the geodesic L is a circle

(c) the geodesic L is an ellipse

(d) the surface S is a sphere

(e) the set of circumstances described is impossible

51. Imagine an infinitely long, stationary, straight line that has always existed, exists at this moment, and always will exist in the future. In 4D Euclidean time-space, the path of this line is

(a) a line

(b) a half-plane

(c) a plane

(d) a sphere

(e) a tesseract

52. Suppose that two straight lines intersect, forming four angles. The two angles opposite each other are called

(a) interior angles

(b) supplementary angles

(c) complementary angles

(d) alternate angles

(e) vertical angles

53. In Fig. Exam-6, suppose the measure of QPR is 50°. The other two interior angles have measures x and y . Which of the following statements can be made with certainty?

(a) x + y = 2 π rad

(b) x + y = π rad

(c) xy = π /2 rad

(d) xy = 50°

(e) x + y = 130°

Fig. Exam-6 . Illustration for Questions 53 and 54 in the final exam.

54. In Fig. Exam-6, suppose the measure of angle y is exactly 90°, the length of side r is exactly 10 meters, and the length of side p is exactly 8 meters. The length of side q is

(a) approximately 1.732 meters

(b) approximately 1.414 meters

(c) exactly 6 meters

(d) exactly 4 meters

(e) impossible to determine without more information

55. Given any three distinct points, they cannot form a triangle if

(a) each one is equidistant from the other two

(b) they lie on different lines

(c) they all lie on the surface of the same sphere

(d) they all lie on the perimeter of the same circle

(e) they all lie on the same line

56. A plane region that does not include its boundary is called

(a) indefinite

(b) non-Euclidean

(c) open

(d) closed

(e) non-contiguous

57. Refer to Fig. Exam-7. The values a, b , and c are

(a) the coordinates of a point on line L

(b) the variables in the equation of line L

(c) the solutions of line L

(d) the direction numbers of line L

(e) none of the above

Fig. Exam-7 . Illustration for Questions 57, 58, and 59 in the final exam.

58. In the situation shown by Fig. Exam-7, suppose the values of a, b , and c are all multiplied by −1. How will vector m and line L be related then?

(a) They will be parallel to each other

(b) They will be perpendicular to each other

(c) They will be skewed with respect to each other

(d) They will intersect each other

(e) It is impossible to say without more information

59. In the scenario of Fig. Exam-7, what can we say about any point ( x,y,z ) on line L ?

(a) Its coordinates are x = x 0 + a, y = y 0 + b , and z = z 0 + c

(b) Its coordinates are x = x 0a, y = y 0b , and z = z 0c

(c) Its coordinates are x = x 0 + at, y = y 0 + bt , and z = z 0 + ct , where t is a real number

(d) Its coordinates are x = tx 0 , y = ty 0 , and z = tz 0 , where t is a real number

(e) None of the above

60. Which of the following (a), (b), (c), or (d), if any, represents the same point on the Cartesian plane as the ordered pair (−0.5, 1.7)?

(a) (−5,17)

(b) (1.7,−0.5)

(c) (−1/2,17/10)

(d) (5,−17)

(e) None of the above

61. Let θ be the measure of an interior angle in a regular polygon. What is the range of possible values for θ ?

(a) 0 rad ≤ θ ≤ 2 π rad

(b) 0 rad < θ < 2 π rad

(c) 0 rad ≤ θπ /2 rad

(d) 0 rad < θ < π /2 rad

(e) None of the above

62. The distance between (2,3,4,5) and (6,7,8,9) in Cartesian 4D hyperspace is

(a) equal to 2

(b) equal to 4

(c) equal to 8

(d) equal to 16

(e) equal to 32

63. Suppose a geometric object in the mathematician’s polar coordinate plane is represented by the equation θ = π /4. The object is

(a) a circle

(b) a hyperbola

(c) a parabola

(d) a straight line

(e) a spiral

64. Imagine a stationary circle that has always existed, exists at this moment, and always will exist in the future. In 4D Euclidean time-space, the path of this circle, not including the points in its interior, is

(a) a hollow, infinitely long cylinder

(b) a solid, infinitely long cylinder

(c) a hollow sphere

(d) a solid sphere

(e) none of the above

65. Refer to Fig. Exam-8. What is the polar equation of the circle shown in this graph?

(a) x 2 + y 2 + a 2

(b) x 2y 2 = a 2

(c) r = a

(d) r = θ

(e) θ = a

Fig. Exam-8 . Illustration for Questions 65, 66, and 67 in the final exam.

66. Refer to Fig. Exam-8. What is the Cartesian-plane equation of the circle shown in this graph?

(a) x 2 + y 2 = a 2

(b) x 2y 2 = a 2

(c) r = a

(d) r = θ

(e) θ = a

67. Refer to Fig. Exam-8. Suppose the radius of the circle is doubled. What is the equation of the resulting circle in polar coordinates?

(a) x 2 + y 2 = 4 a 2

(b) x 2y 2 = 4 a 2

(c) r = 2 a

(d) r = 2 θ

(e) θ = 2 a

68. Suppose a line segment is bisected, and then the resulting line segments are bisected, and then the resulting line segments are bisected. How many times can this process, in theory, be repeated?

(a) Until the bisection produces individual points

(b) Forever

(c) It depends on the length of the initial line segment

(d) It depends on whether or not the end points are considered part of the initial line segment

(e) It depends on whether or not the initial line segment is straight

69. Suppose there are two spheres S 1 and S 2 , and the radius of S 2 is exactly four times the radius of S 1 . Suppose A 1 is the surface area of S 1 , and A 2 is the surface area of S 2 . Which, if any, of the following equations (a), (b), (c), or (d) is true?

(a) A 2 = 4 A 1

(b) A 2 = 8 A 1

(c) A 2 = 16 A 1

(d) A 2 = 32 A 1

(e) None of the above equations (a), (b), (c), or (d) is true

70. Suppose X is a plane, and P is a point. How many lines can exist that are normal to plane X and that pass through P ?

(a) None

(b) One

(c) Two

(d) Infinitely many

(e) It depends on whether or not P is in plane X

71. In a right circular cone, a line segment that connects the apex (top) and the center of the base

(a) has a length equal to the radius of the base

(b) has a length equal to half the radius of the base

(c) has a length equal to twice the radius of the base

(d) has a length equal to the circumference of the base

(e) is perpendicular to the plane containing the base

72. Suppose you draw a line L and a point P near that line. Then you drop a perpendicular from point P to line L , and let Q be the point where the perpendicular intersects the line. Then you draw a point R on line L , different from point Q . The points P, Q, and R lie at the vertices of

(a) a right triangle

(b) an equilateral triangle

(c) an isosceles triangle

(d) a congruent triangle

(e) none of the above

73. The distance between two parallel planes

(a) is expressed along lines contained within both planes

(b) is expressed along lines normal to both planes

(c) is expressed along lines that intersect neither plane

(d) varies with location

(e) cannot be defined

74. Let L be a line parallel to a plane X . How many lines can exist in plane X that are skew to line L ?

(a) None

(b) One

(c) Two

(d) Three

(e) Infinitely many

75. A pyramid with a square base has

(a) four faces in all

(b) five faces in all

(c) four or five faces in all

(d) slant faces that are all rectangles

(e) faces that are all congruent

76. If rotational sense is an important consideration in the expression of an angle θ , the counterclockwise sense usually indicates

(a) θ = 0°

(b) θ < 0°

(c) θ > 0°

(d) − π rad < θ < π rad

(e) −180° < θ < 180°

77. In the dimensionally reduced illustration Fig. Exam-9, the pitch of the cone (that is, the angle between the cone surface and the + t axis) depicts

(a) the path of a single photon through time-space

(b) the hyperspace locations of the photons that came from the bulb at the instant it was switched on

(c) the hyperspace locations of all the photons that have come from the bulb since the instant it was switched on

(d) the speed of light

(e) the rate at which the observer travels through time

78. In the dimensionally reduced illustration Fig. Exam-9, imagine some plane X that is parallel to the xy -plane and that passes through the cone, so the set of points representing the intersection between plane X and the cone surface (not including the interior of the cone) is a circle. This circle represents

(a) the path of a single photon through time-space

(b) the hyperspace locations of the photons that came from the bulb at the instant it was switched on

(c) the hyperspace locations of all the photons that have come from the bulb since the instant it was switched on

(d) the speed of light

(e) the rate at which the observer travels through time

Fig. Exam-9 . Illustration for Questions 77, 78, and 79 in the final exam.

79. In the dimensionally reduced illustration Fig. Exam-9, imagine some plane X that is parallel to the xy -plane and that passes through the cone, so the set of points representing the intersection between plane X and the cone (including the interior of the cone) is a disk. This disk represents

(a) the path of a single photon through time-space

(b) the hyperspace locations of the photons that came from the bulb at the instant it was switched on

(c) the hyperspace locations of all the photons that have come from the bulb since the instant it was switched on

(d) the speed of light

(e) the rate at which the observer travels through time

80. Which of the following equations represents a parabola in Cartesian coordinates?

(a) y = 3 x

(b) x 2 + y 2 = 1

(c) x 2y 2 = 1

(d) y = 3 x 2 + 2 x − 5

(e) x = −2 y + 5

81. A triangle cannot be both

(a) isosceles and equilateral

(b) isosceles and right

(c) acute and obtuse

(d) Euclidean and equilateral

(e) Eucidean and isosceles

82. An uncalibrated drafting compass and a pencil, without a straight edge, can be used to

(a) construct a line segment connecting two defined points

(b) construct a line segment passing through a single defined point

(c) construct an arc centered at a defined point

(d) construct a triangle connecting three defined points

(e) none of the above

83. An uncalibrated straight edge and a pencil, without a compass, can be used to

(a) drop a perpendicular to a line from a defined point not on that line

(b) construct an arc centered at a defined point

(c) construct an arc passing through a defined point

(d) construct a parallel to a line, passing through a defined point not on that line

(e) none of the above

84. Imagine a triangle with interior angles measuring 30°, 60°, and 100°. What can be said about this triangle?

(a) It must be a right triangle

(b) It must be a non-Euclidean triangle

(c) It must be an isosceles triangle

(d) It must be a congruent triangle

(e) It must be an acute triangle

85. Consider an arc of a circle measuring 2 radians. Suppose the radius of the circle is 1 meter. What is the area of the circular sector defined by this arc?

(a) 1/2 square meter

(b) 1 square meter

(c) 1/ π square meter

(d) 2/ π square meter

(e) π square meters

86. Suppose that a straight section of railroad crosses a straight stretch of highway. The acute angle between the tracks and the highway center line measures exactly 70°. What is the measure of the obtuse angle between the tracks and the highway center line?

(b) 70°

(c) 90°

(d) 110°

(e) 290°

87. Suppose we are told two things about a quadrilateral: first, that it is a rhombus, and second, that one of its interior angles measures 70°. The measure of the angle adjacent to the 70° angle is

(a) 20°

(b) 70°

(c) 90°

(d) 150°

(e) none of the above

88. Suppose the coordinates of a point in the mathematician’s polar plane are specified as ( θ,r ) = (− π /4,2). This is equivalent to the coordinates

(a) ( π /4,2)

(b) (3 π /4,2)

(c) (5 π /4,2)

(d) (7 π /4,2)

(e) none of the above

89. Suppose the cylindrical coordinates of a certain object in the sky are specified as ( θ,r,h ), where θ is its azimuth as expressed in the plane of the horizon, r is its horizontal distance from us (also called its distance downrange), and h is its altitude with respect to the plane of the horizon. Imagine that the object flies directly away from us, so r is doubled but θ remains constant. What happens to h ?

(a) It does not change

(b) It doubles

(c) It becomes four times as great

(d) If becomes half as great

(e) It becomes one-quarter as great

90. Suppose the cylindrical coordinates of a certain object in the sky are specified as ( θ,r,h ), where θ is its azimuth as expressed in the plane of the horizon, r is its horizontal distance from us (also called its distance downrange), and h is its altitude with respect to the plane of the horizon. Imagine that the object flies straight up vertically into space, perpendicular to the plane containing the horizon, so h increases without limit. What happens to θ and r ?

(a) Both θ and r remain unchanged

(b) θ approaches 90°, while r increases without limit

(c) θ remains unchanged, while r increases without limit

(d) θ increases without limit, while r remains unchanged

(e) It is impossible to answer this without more information

91. What is the slope m of the graph of the equation y = 3 x 2 ?

(a) m = 3

(b) m = −3

(c) m = 1/3

(d) m = −1/3

(e) None of the above

92. Suppose we are told that a plane quadrilateral has diagonals that bisect each other. We can be certain that this quadrilateral is

(a) a square

(b) a rhombus

(c) a rectangle

(d) a parallelogram

(e) irregular

93. Suppose L is a line and P is a point not on L . Then there is one, but only one, line M through P , such that M is parallel to L. This statement is an axiom that holds true on

(a) the surface of a flat plane

(b) the surface of a sphere

(c) any surface with positive curvature

(d) any surface with negative curvature

(e) any surface

94. Refer to Fig. Exam-10. Suppose two rays intersect at point P (drawing A). You set down the non-marking tip of a compass on P , and construct an arc from one ray to the other, creating intersection points R and Q (drawing B). Then, you place the non-marking tip of the compass on Q , increase its span somewhat from the setting used to generate arc QR , and construct a new arc. Next, without changing the span of the compass, you set its non-marking tip on R and construct an arc that intersects the arc centered at Q . Let S be the point at which the two arcs intersect (drawing C). Finally, you construct ray PS , as shown at D. Which of the following statements (a), (b), (c), or (d) is true?

(a) RPS and SPQ have equal measure

(b) PQS is a right angle

(c) PRS is a right angle

(d) Line segment PS is twice as long as line segment RQ

(e) None of the above statements (a), (b), (c), or (d) is true

Fig. Exam-10 . Illustration for Questions 94, 95, and 96 in the final exam.

95. In the situation shown by Fig. Exam-10, and according to the description given in the previous question, which of the following statements is true?

(a) Points R, P , and Q lie at the vertices of an isosceles triangle

(b) Points R, P , and S lie at the vertices of a right triangle

(c) Points R, P , and Q lie at the vertices of an equilateral triangle

(d) Quadrilateral RPQS is a trapezoid

(e) Quadrilateral RPQS is a parallelogram

96. In the situation shown by Fig. Exam-10, and according to the description given in the previous question, consider the triangle whose vertices are points R, P , and S . Also consider the triangle whose vertices are points P, Q , and S . These two triangles

(a) are directly congruent

(b) are inversely congruent

(c) are both right triangles

(d) are both isosceles triangles

(e) do not resemble each other in any particular way

97. Refer to Fig. Exam-11. The perimeter, B , of quadrilateral PQRS is given by which of the following formulas?

(a) B = 2 d + 2 e

(b) B = ed

(c) B = 2 f + 2 g

(d) B = fg

(e) None of the above

98. Refer to Fig. Exam-11. Suppose line segments PQ and SR are parallel to each other, and the line segment whose length is m bisects both line segments PS and QR . Based on this information, which of the following is true?

(a) m = fd /2

(b) m = ( f + d )/2

(c) m = eg /2

(d) m = ( e + g )/2

(e) None of the above

99. Refer to Fig. Exam-11. Suppose line segments PQ and SR are parallel to each other, and the line segment whose length is m bisects both line segments PS and QR . Based on this information, which of the following statements (a), (b), (c), or (d) is not necessarily true?

(a) Quadrilateral PQRS is a trapezoid

(b) The line segment whose length is m is parallel to line segments PQ and SR

(c) The distances e and g are equal

(d) The distance h cannot be greater than the distance e or the distance g

(e) All of the statements (a), (b), (c), and (d) are true

Fig. Exam-11 . Illustration for Questions 97, 98, and 99 in the final exam.

100. Which of the following criteria can be used to establish the fact that two triangles are directly congruent?

(a) All three corresponding sides must have equal lengths

(b) All three corresponding angles must have equal measures

(c) The Pythagorean theorem must hold for both triangles

(d) Both triangles must be right triangles

(e) Both triangles must have the same perimeter

1. d

2. c

3. b

4. b

5. c

6. b

7. a

8. c

9. b

10. c

11. e

12. a

13. a

14. d

15. b

16. c

17. c

18. a

19. d

20. c

21. b

22. c

23. c

24. d

25. b

26. a

27. b

28. c

29. c

30. e

31. c

32. c

33. e

34. b

35. c

36. e

37. e

38. b

39. c

40. d

41. a

42. c

43. a

44. c

45. b

46. b

47. c

48. d

49. e

50. a

51. c

52. e

53. e

54. c

55. e

56. c

57. d

58. a

59. c

60. c

61. e

62. c

63. d

64. a

65. c

66. a

67. c

68. b

69. c

70. b

71. e

72. a

73. b

74. e

75. b

76. c

77. d

78. b

79. c

80. d

81. c

82. c

83. e

84. b

85. b

86. d

87. e

88. d

89. b

90. a

91. e

92. d

93. a

94. a

95. a

96. b

97. e

98. b

99. c

100. a

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