Download ACT Geometry Practice - Ms-Schmitz-Geometry

ACT Geometry Practice
Multiple Choice
Identify the letter of the choice that best completes the statement or answers the question.
Refer to Figure 1.
Figure 1
____
____
1. Which is these is NOT a way to refer to line BD?
a.
c.
b. m
d. line JD
2. Are points A, C, D and F coplanar? Explain.
a.
b.
c.
d.
Yes; they all lie on plane P.
No; they are not on the same line.
Yes; they all lie on the same face of the pyramid.
No; three lie on the same face of the pyramid and the fourth does not.
Refer to Figure 2.
Figure 2
____
____
3. Name a point that is NOT coplanar with G, A, and B.
a. K
c. C
b. D
d. F
4. Name an intersection of plane GFL and the plane that contains points A and C.
a. line LC
c. line AC
b. C
d. plane CAB
In the figure,
____
5. If
a. 33
b. 58
bisects
and
.
, find x.
c. 11
d. 29
In the figure,
____
____
____
and
are opposite rays.
6. Which is NOT true about
?
a.
is acute.
b.
c. Point M lies in the interior of
d. It is an angle bisector.
7. If
and
a. 137
b. 12
8.
. Point R lies on
and
.
.
, what is
?
c. 4.2
d. 43
. If
a. 11.5
b. 7.83
bisects
, find f.
c. 3.5
d. 19
Name each polygon by its number of sides. Then classify it as convex or concave and regular or
irregular.
____
9.
a. triangle, convex, regular
b. triangle, concave, irregular
c. triangle, convex, irregular
d. quadrilateral, convex, irregular
Find the length of each side of the polygon for the given perimeter.
____
10.
in. Find the length of each side.
____
a. 11 in., 20 in., 35 in.
b. 10 in., 18.5 in., 31.5 in.
11.
cm. Find the length of each side.
a. 8 cm, 8 cm, 8 cm
b. 6 cm, 6 cm, 6 cm
c. 12 in., 21.5 in., 38.5 in.
d. 10 in., 15 in., 35 in.
c. 10 cm, 10 cm, 4 cm
d. 9 cm, 9 cm, 6 cm
Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.
____
12. Given: point B is in the interior of
.
Conjecture:
a. False;
may be obtuse.
b. True
c. False; just because it is in the interior does not mean it is on the bisecting line.
d. False;
.
____
13. Given:
Conjecture:
a. False;
.
c. False;
b. True
d. False;
14. Given: Two angles are supplementary.
Conjecture: They are both acute angles.
a. False; either both are right or they are adjacent.
b. True
c. False; either both are right or one is obtuse.
d. False; they must be vertical angles.
____
.
.
____
15. Given: segments RT and ST; twice the measure of
Conjecture: S is the midpoint of segment RT.
a. True
b. False; point S may not be on .
c. False; lines do not have midpoints.
d. False;
could be the segment bisector of .
Refer to the figure below.
____
____
____
16. Name all planes intersecting plane CDI.
a. ABC, CBG, ADI, FGH
b. CBA, DAF, HGF
c. BAD, GFI, CBG, GFA
d. DAB, CBG, FAD
17. Name all segments parallel to
a.
b.
c.
d.
18. Name all segments skew to
a.
b.
.
.
c.
d.
.
____
19. In the figure,
. Find x and y.
a.
b.
c.
d.
Given the following information, determine which lines, if any, are parallel. State the postulate or
theorem that justifies your answer.
____
20.
a.
b.
c.
d.
; congruent corresponding angles
; congruent corresponding angles
; congruent alternate interior angles
; congruent alternate interior angles
Find each measure.
____
21.
a.
b.
c.
d.
Name the congruent angles and sides for the pair of congruent triangles.
____
22.
a.
b.
c.
d.
Determine whether
given the coordinates of the vertices. Explain.
____
23.
____
a. Yes; two sides of triangle PQR and angle PQR are the same measure as the
corresponding sides and angle of triangle STU.
b. Yes; each side of triangle PQR is the same length as the corresponding side of
triangle STU.
c. No; one of the triangles is obtuse.
d. No; each side of triangle PQR is not the same length as the corresponding side of
triangle STU.
24. Triangles ABC and AFD are vertical congruent equilateral triangles. Find x and y.
a.
c.
b.
d.
____
25. Triangles MNP and OMN are congruent equilateral triangles. Find x and y.
a.
b.
____
26.
c.
d.
is an altitude.
a. 34
b. 32
,
. Find
.
c. 18
d. 31
Determine the relationship between the lengths of the given sides.
____
27.
a.
b.
c. cannot be determined
d.
Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or
no. Explain.
____
28. 3, 9, 10
a. Yes; the third side is the longest.
b. No; the sum of the lengths of two sides is not greater than the third.
c. No; the first side is not long enough.
d. Yes; the sum of the lengths of any two sides is greater than the third.
____
29. An isosceles triangle has a base 9.6 units long. If the congruent side lengths have measures to the
first decimal place, what is the shortest possible length of the sides?
a. 4.9
c. 4.7
b. 19.3
d. 9.7
Determine whether each pair of figures is similar. Justify your answer.
____
30.
a.
b.
is not similar to
. Corresponding angles are not the same.
because the corresponding angles of each triangle are congruent.
The ratio of the corresponding sides is 1.
c.
is not similar to
. The ratios of the corresponding sides are not the
same.
d.
because the corresponding angles of each triangle are congruent.
The ratio of the corresponding sides is 2.
Find x and the measures of the indicated parts.
____
31.
AB and BC
a.
b.
c.
d.
____
32.
BD and CE
a.
c.
b.
d.
Find the perimeter of the given triangle. Round your answer to the nearest tenth if necessary.
____
33.
, if
,
is a parallelogram,
a. 32
b. 56
____
____
34. Find AD if
,
,
, and
.
c. 24
d. 13.7
,
and
are medians,
,
,
, and
.
a. 15
c. 6
b. 48
d. 18
35. Dante is standing at horizontal ground level with the base of the Empire State Building in New
York City. The angle formed by the ground and the line segment from his position to the top of the
building is 48.4°. The height of the Empire State Building is 1472 feet. Find his distance from the
Empire State Building to the nearest foot.
a. 1307
c. 2217
b. 7.65
d. 1968
____
36. A space shuttle is one kilometer above sea level when it begins to climb at a constant angle of 3°
for the next 80 ground kilometers. About how far above sea level is the space shuttle after its
climb?
a. 4.2 kilometers
c. 79.9 kilometers
b. 5.2 kilometers
d. 80.9 kilometers
A landscaper is making a retaining wall to shore up the side of a hill. To ensure against collapse,
the wall should make an angle 75° or less with the ground.
____
37. How far from the base of the hill is the base of a 15-foot slanted wall?
a. 3.88 ft
c. 55.98 ft
b. 14.49 ft
d. 57.96 ft
A 60-yard long drawbridge has one end at ground level. The other end is initially at an incline of
5°.
____
____
____
38. During one stage of the drawbridge’s motion, the raised end is 15 yards above the ground. What is
the incline of the drawbridge to the nearest hundredth?
a. 0.004°
c. 14.48°
b. 14.04°
d. 75.52°
39. Two horses are observed by a hang glider 80 meters above a meadow. The angles of depression
are 10.4° and 8°. How far apart are the horses?
a. 133.3 m
c. 569.2 m
b. 435.9 m
d. 1005.1 m
40. Zack, Rachel, and Maddie are unraveling a huge ball of yarn to see how long it is. As they move
away from each other, they form a triangle. The distance from Zack to Rachel is 3 meters. The
distance from Rachel to Maddie is 2.5 meters. The distance from Maddie to Zack is 4 meters. Find
the measures of the three angles in the triangle.
a.
,
,
b.
,
,
c.
,
,
d.
,
,
Complete the statement about parallelogram ABCD.
____
41.
a.
b.
c.
d.
____
; Alternate interior angles are congruent.
; Alternate interior angles are congruent.
; Opposite angles of parallelograms are congruent.
; Opposite angles of parallelograms are congruent.
Determine whether a figure with the given vertices is a parallelogram. Use the method indicated.
____
42.
a.
b.
c.
d.
____
,
,
,
; Distance and Slope Formulas
no; The opposite sides are not congruent and do not have the same slope.
yes; The opposite sides do not have the same slope.
no; The opposite sides do not have the same slope.
yes; The opposite sides are not congruent and do not have the same slope.
43.
In rhombus YZAB, if
a. 24
b. 12
12, find
.
c. 6
d.
Given each set of vertices, determine whether parallelogram ABCD is a rhombus, a rectangle, or a
square. List all that apply.
____
44.
,
,
,
a. square; rectangle; rhombus
b. rhombus
c. square
d. rectangle
____
45. For trapezoid JKLM, A and B are midpoints of the legs. Find ML.
____
a. 4
c. 68
b. 34
d. 40
46. Find the exact circumference of the circle.
a. 12
b. 24
mm
mm
c. 12 mm
d. 6
mm
Use the diagram to find the measure of the given angle.
____
47.
a. 50
b. 40
c. 60
d. 30
____
48. In
, TS = 15, UQ = US. Find m
a. 28
b. 30
____
.
c. 15
d. 39
49.
If
a. 40
b. 25
= 40,
= 120, and
= 110, find
c. 50
d. 80
.
____
50. Find x. Assume that segments that appear tangent are tangent.
a. 7
b. 6
c. 14
d. 5
Find the measure of the numbered angle.
____
51.
a. 62.5
b. 105
c. 112.5
d. 115
Find x. Assume that any segment that appears to be tangent is tangent.
____
____
52.
a. 12
b. 14
c. 8
d. 10
a. 35
b. 20
c. 25
d. 30
53.
Find x. Round to the nearest tenth if necessary.
____
54.
a. 7
b. 8
c. 9
d. 10
Find x. Round to the nearest tenth if necessary. Assume that segments that appear to be tangent
are tangent.
____
55.
a. 5
b. 2.8
c. 3.5
d. 4
____
56.
____
a. 3
c. 2
b. 7
d. 4
57. Find the perimeter and area of the parallelogram. Round to the nearest tenth if necessary.
____
a. 88 mm; 415.7
c. 44 mm; 415.7
b. 44 mm; 346.4
d. 88 mm; 346.4
58. Find the area of the figure. Round to the nearest tenth if necessary.
a. 672
b. 74
c. 336
d. 1344
Find the area of the figure. Round to the nearest tenth if necessary.
____
59.
a. 148 units2
b. 140 units2
____
____
c. 130 units2
d. 122 units2
60.
a. 541.9 units2
c. 192 units2
2
b. 624 units
d. 315.8 units2
61. The solid below is a composite of a cube and a square pyramid. The base of the solid is the base of
the cube. Find the lateral area of the solid.
a. 80
b. 660
c. 560
d. 720
Find the surface area of the regular pyramid. Round to the nearest tenth if necessary.
____
62.
a. 496.7
b. 5184.0
____
c. 864.0
d. 356.4
63. Find the lateral area of the cone. Use 3.14 for . Round to the nearest tenth if necessary.
a. 35.0
b. 109.9
c. 188.4
d. 183.2
Determine whether each statement is true or false. If false, give a counterexample.
____
64. If a plane intersects a sphere so that it contains the center of the sphere, then that intersection will
sometimes be a great circle.
a. True
b. False, the intersection will never be a great circle.
c. False, the intersection will always be a great circle.
d. False, the intersection of a plane and a sphere cannot contain the center of the
sphere.
____
____
____
____
____
65. All radii of the same sphere are congruent.
a. True
b. False, no two radii of the sphere are congruent.
c. False, some radii will be congruent, but not all.
d. False, a segment joining two points on the sphere is a tangent.
66. Find the surface area of the sphere. Use 3.14 for . Round to the nearest tenth.
a. 113
c. 28.3
b. 339.1
d. 36
67. Find the surface area of a sphere if the circumference of a great circle is 43.96 centimeters. Use
3.14 for . Round to the nearest tenth.
a. 4308.1
c. 153.9
b. 196
d. 615.4
68. Suppose a snow cone has a paper cone that is 8 centimeters deep and has a diameter of 5
centimeters. The flavored ice comes in a spherical scoop with a diameter of 5 centimeters and rests
on top of the cone. If all the ice melts into the cone, will the cone overflow? Explain.
a. No. The volume of the ice is less than the volume of the cone.
b. No. The volume of the ice is exactly the same as the volume of the cone.
c. Yes. The volume of the ice is greater than the volume of the cone.
d. There is not enough information given to solve this problem.
69. What is the volume of this sphere, rounded to the nearest tenth?
a. 7234.6 cm3
c. 602.9 cm3
3
b. 1808.6 cm
d. 150.7 cm3
Determine whether each statement is sometimes, always, or never true.
____
70. Congruent spheres have equal surface areas.
a. sometimes
c. never
b. always
ACT Geometry Practice
Answer Section
MULTIPLE CHOICE
1. ANS: C
The proper way to refer to a line is any 2 points on the line with an arrow above them or “line
such-and-such”, where “such-and-such” is any 2 points on the line. Using three letters is not
correct.
TOP: Identify and model points¸ lines¸ and planes.
KEY: Points, Lines,
Planes
NOT: /A/ Does line BD contain point J? /B/ Does that line contain points B and D? /C/ Correct!
/D/ Are points J and D on line BD?
2. ANS: D
Points that lie on the same plane are said to be coplanar. Three points are always coplanar but if
the fourth point is not on the same plane with the first three, they are not all coplanar.
TOP: Identify coplanar points and intersecting lines in space.
KEY: Coplanar Points, Intersecting Lines, Lines in Space
NOT: /A/ Do all four points lie on the same plane? Which plane? /B/ Do all four points lie on the
same plane? Which plane?/C/ What does coplanar mean? /D/ Correct!
3. ANS: C
Coplanar points are points that lie on the same plane.
TOP: Identify planes in space.
KEY: Planes, Planes in Space
NOT: /A/ Is K in a different plane? /B/ What plane are you working with? /C/ Correct! /D/ What
plane are you working with?
4. ANS: A
The intersection of two planes is a line.
TOP: Identify planes in space.
KEY: Planes, Planes in Space
NOT: /A/ Correct! /B/ Can the intersection of two planes be a point? /C/ Is point A on plane
GFL? /D/ Can the intersection of two planes be a plane?
5. ANS: D
Since
bisects
the equation to find x.
,
and
. Solve for v, then substitute into either side of
TOP: Identify and use congruent angles. KEY: Angles, Congruent Angles, Congruency
NOT: /A/ Don’t forget to subtract./B/ You are not finding the measure of FGH. You are finding
x. /C/ You are not finding v. You are finding x. /D/ Correct!
6.
ANS:
A
so it is obtuse.
TOP: Identify and use the bisector of an angle.
KEY: Angle Bisectors
NOT: /A/ Correct! /B/ If answer d is true, then this must be true. /C/ Being in the interior means
being between the two end rays of an angle. /D/ If answer b is true, then this must be true.
7. ANS: D
TOP: Identify and use the bisector of an angle.
KEY: Angle Bisectors
NOT: /A/ What angle are you looking for? /B/ Did you solve for q instead of the angle measure?
/C/ What two angles added together equal angle LKN? /D/ Correct!
8. ANS: A
Lines that form right angles are perpendicular. A right angle measures 90.
TOP: Identify perpendicular lines.
KEY: Perpendicular Lines
NOT: /A/ Correct! /B/ Did you use the Distributive Property carefully? /C/ Check your math. /D/
What is the measure of the angle created by perpendicular lines?
9. ANS: C
Suppose the line containing each side is drawn. If any of the lines contain any point in the interior
of the polygon, then it is concave. Otherwise it is convex.
A convex polygon in which all the sides are congruent and all the angles are congruent is called a
regular polygon.
TOP: Name polygons.
KEY: Polygons, Name Polygons
NOT: /A/ If it is regular the angles and sides would all be congruent. /B/ If it is concave lines
drawn from the segments would pass through the polygon. /C/ Correct! /D/ Count the number of
sides.
10. ANS: B
Perimeter is the sum of the sides.
TOP: Find the perimeters of polygons.
KEY: Perimeter, Polygons
NOT: /A/ What is the sum of the sides? /B/ Correct! /C/ Did you find the value of y? /D/ What is
the value of y?
11. ANS: A
Perimeter is the sum of the sides.
TOP: Find the perimeters of polygons.
KEY: Perimeter, Polygons
NOT: /A/ Correct! /B/ How many sides does the figure have? /C/ Should all the sides have the
same length? /D/ The sides are congruent.
12. ANS: C
Angles are congruent only if their measures are equal. Point B may be closer to line AD or line DC
so the measures would not be equal.
TOP: Find counterexamples.
KEY: Counterexamples
NOT: /A/ What is the definition of congruent? /B/ What is the definition of congruent? /C/
Correct! /D/ Would that be a counterexample?
13. ANS: D
Because m is squared in the example, m could be positive or negative.
TOP: Find counterexamples.
KEY: Counterexamples
NOT: /A/ Subtract 6 from both sides. /B/ What about negative numbers? /C/ Subtract 6 from both
sides. /D/ Correct!
14. ANS: C
If two angles are supplementary their measures total 180. Either both are right or one is obtuse and
the other acute.
TOP: Find counterexamples.
KEY: Counterexamples
NOT: /A/ What is the definition of supplementary? /B/ What is the definition of supplementary?
/C/ Correct! /D/ What is the definition of supplementary?
15. ANS: B
Even though they have a common point, the two segments do not have to be on the same line.
TOP: Find counterexamples.
KEY: Counterexamples
NOT: /A/ What is the definition of midpoint? /B/ Correct! /C/ What is the definition of midpoint?
/D/ What is the definition of midpoint?
16. ANS: A
Planes that intersect have a common line.
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines, Relationship Between Two Planes
NOT: /A/ Correct! /B/ This plane has four lines to intersect with other planes. /C/ Do they all
intersect CDI in a line? /D/ This plane has four lines to intersect with other planes.
17. ANS: B
Coplanar segments that do not intersect are parallel.
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines, Relationship Between Two Planes
NOT: /A/ Are those parallel to GF? /B/ Correct! /C/ Is that all? /D/ Is that all?
18. ANS: D
Skew lines do not intersect and are not coplanar.
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines, Relationship Between Two Planes
NOT: /A/ Are any of those segments in the same plane as segment BC? /B/ Skew lines are not
coplanar. /C/ Do any of those segments intersect segment BC? /D/ Correct!
19. ANS: C
Corresponding angles are congruent.
Alternate interior angles are congruent.
Consecutive interior angles are supplementary.
Alternate exterior angles are congruent.
TOP: Use algebra to find angle measures. KEY: Angles, Angle Measures
NOT: /A/ What do supplementary angles add up to?/B/ What do the angles of a right triangle add
up to? /C/ Correct! /D/ Is that triangle isosceles?
20. ANS: C
Postulates and theorems:
If corresponding angles are congruent, then lines are parallel.
If given a line and a point not on the line, then there exists exactly one line through the point that is
parallel to the given line.
If alternate exterior angles are congruent, then lines are parallel.
If consecutive interior angles are supplementary, then lines are parallel.
If alternate interior angles are congruent, then lines are parallel.
If 2 lines are perpendicular to the same line, then lines are parallel.
TOP: Recognize angle conditions that occur with parallel lines. KEY: Angles, Parallel Lines
NOT: /A/ What kind of angles are those? /B/ What kind of angles are those?/C/ Correct! /D/
Which lines are parallel?
21. ANS: A
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to
the sum of the measures of the two remote interior angles.
TOP: Apply the Exterior Angle Theorem. KEY: Exterior Angle Theorem
NOT: /A/ Correct! /B/ What is the sum of the measures of the angles in a triangle? /C/ Did you
use the Exterior Angle Theorem?/D/ Use the Exterior Angle Theorem.
22. ANS: C
The corresponding sides and angles can be determined from any congruence statement by
following the order of the letters.
TOP: Name and label corresponding parts of congruent triangles.
KEY: Corresponding Parts, Congruent Triangles
NOT: /A/ Did you follow the order of the letters?/B/ The corresponding sides and angles can be
determined from any congruence statement by following the order of the letters. /C/ Correct! /D/
Did you follow the order of the letters?
23. ANS: D
If each side of triangle PQR is the same length as the corresponding side of triangle STU, then the
triangles are congruent.
TOP: Use the SSS Postulate to test for triangle congruence.
KEY: SSS Postulate, Congruent Triangles
NOT: /A/ Use the SSS Postulate. /B/ Check your math. /C/ How do you determine if two triangles
are congruent? /D/ Correct!
24. ANS: A
TOP: Use properties of equilateral triangles.
KEY: Equilateral Triangles
NOT: /A/ Correct! /B/ What do you know about the sides of an equilateral triangle? /C/ How
many degrees is each angle of an equilateral triangle? /D/ Did you add or subtract when solving for
y?
25. ANS: A
TOP: Use properties of equilateral triangles.
KEY: Equilateral Triangles
NOT: /A/ Correct! /B/ Did you set the sides equal to each other? /C/ How many degrees is each
angle of an equilateral triangle? /D/ Check your math.
26. ANS: A
If
is an altitude,
. Also, the measures of the angles of every triangle add up to 180.
TOP: Identify and use altitudes in triangles.
KEY: Altitudes,
Triangles
NOT: /A/ Correct! /B/ Which angle measures add up to 180?/C/ Which angles must measure 90°?
/D/ Check your math.
27. ANS: C
The measures of the angles opposite the sides given are compared. The larger the angle, the longer
its side.
TOP: Recognize and apply properties of inequalities to the relationships between angles and
sides of a triangle.
KEY: Properties of Inequality, Triangles
NOT: /A/ Check the angles opposite the sides. /B/ Are the angles in the same or congruent
triangles? /C/Correct!/D/ Check the angles opposite the sides.
28. ANS: D
The sum of the lengths of any two sides must be greater than the third.
TOP: Apply the Triangle Inequality Theorem.
KEY: Triangles Inequality Theorem
NOT: /A/ Did you check all the sums? /B/ Add two sides and compare to the third. /C/ Add two
sides and compare to the third./D/ Correct!
29. ANS: A
The sum of the lengths of any two sides must be greater than the third.
TOP: Determine the shortest distance between a point and a line.
KEY: Distance, Distance Between a Point and a Line
NOT: /A/ Correct! /B/ Would both sides have to be longer than the base?/C/ Is the sum of the two
sides longer than the base? /D/ Is that the shortest possible length?
30. ANS: B
Two polygons are similar if and only if their corresponding angles are congruent and the measures
of their corresponding sides are proportional.
TOP: Identify similar figures.
KEY: Similar Figures
NOT: /A/ Check the size of the angles./B/ Correct! /C/ Determine the ratio of the corresponding
sides. /D/ Check the similarity statement.
31. ANS: A
Determine the ratio of corresponding parts. Use the ratio to find the missing information.
TOP: Use similar triangles to solve problems.
KEY: Similar Triangles, Solve Problems
NOT: /A/ Correct! /B/ Check your ratio. /C/ What are the values of AB and BC? /D/ Check your
addition.
32. ANS: A
Determine the ratio of corresponding parts. Use the ratio to find the missing information.
TOP: Use similar triangles to solve problems.
KEY: Similar Triangles, Solve Problems
NOT: /A/ Correct! /B/ Check your addition. /C/ Check the ratios. /D/ Check the ratios and your
multiplication.
33. ANS: A
Use the Proportional Perimeter Theorem to find the perimeter of the selected triangle.
TOP: Recognize and use proportional relationships of corresponding perimeters and altitudes.
KEY: Corresponding Perimeters, Corresponding Altitudes
NOT: /A/ Correct! /B/ This is the perimeter of the wrong triangle. /C/ This is the difference
between the perimeters of the two similar triangles. /D/ Remember to find all sides of the triangle.
34. ANS: D
Write a proportion. Substitute the given values for the sides. Cross multiply, and simplify.
TOP: Recognize and use proportional relationships of corresponding angle bisectors and medians
of similar triangles. KEY: Corresponding Angle Bisectors, Corresponding Medians
NOT: /A/ This is the value of the variable. /B/ Which side does the question ask for? /C/ This is
the ratio of the corresponding parts. /D/ Correct!
35. ANS: A
Draw a picture of the situation. Determine which trigonometric ratio should be used to solve.
Substitute the numbers given. Solve for the answer.
TOP: Solve problems using trigonometric ratios.
KEY: Trigonometric Ratios, Solve Problems
NOT: /A/ Correct! /B/ Check the trigonometric ratio. /C/ Which trigonometric ratio should be
used? /D/ Which trigonometric ratio should be used?
36. ANS: B
Draw a picture of the situation. Determine which trigonometric ratio should be used to solve.
Substitute the numbers given. Solve for the answer.
TOP: Solve problems using trigonometric ratios.
KEY: Trigonometric Ratios, Solve Problems
NOT: /A/ Remember to include the initial height of one kilometer. /B/ Correct! /C/ Which
trigonometric ratio should be used? /D/ Which trigonometric ratio should be used?
37. ANS: A
Draw a picture of the situation. Determine which trigonometric ratio should be used to solve.
Substitute the numbers given. Solve for the answer.
TOP: Solve problems using trigonometric ratios.
KEY: Trigonometric Ratios, Solve Problems
NOT: /A/ Correct! /B/ Do not use the sine ratio. /C/ Do not use the tangent ratio. /D/ What is the
cosine ratio?
38. ANS: C
Draw a picture of the situation. Determine which trigonometric ratio should be used to solve.
Substitute the numbers given. Solve for the answer.
TOP: Solve problems involving angles of elevation.
KEY: Angle of
Elevation
NOT: /A/ Did you use the inverse sine to solve. /B/ Do not use the tangent ratio. /C/ Correct! /D/
Do not use the cosine ratio.
39. ANS: A
Draw a picture of the situation. Determine which trigonometric ratio should be used to solve.
Substitute the numbers given. Solve for the answer.
TOP: Solve problems involving angles of depression.
KEY: Angle of Depression
NOT: /A/ Correct! /B/ This is the horizontal distance from one horse to the hang glider. /C/ This
is the horizontal distance from one horse to the hang glider. /D/ Subtract rather than add the
distances.
40. ANS: A
Substitute the given values into the Law of Cosines. Simplify the equation. Find the measure of the
stated angle by using inverse cosine.
TOP: Solve problems by using the Law of Cosines.
KEY: Law of Cosines, Solve Problems
NOT: /A/ Correct! /B/ Which angle goes with each vertex? /C/ Which angle goes with each
vertex? /D/ The triangle is not equilateral.
41. ANS: A
Locate the angle on the parallelogram. Using the properties of parallelograms, determine which
angle is congruent to that angle.
TOP: Recognize and apply properties of the sides and angles of parallelograms.
KEY: Parallelograms, Properties of Parallelograms
NOT: /A/ Why are these angles congruent? /B/ Check the angle and reason. /C/ Correct! /D/ This
angle is not congruent to the original angle.
42. ANS: A
Using the method indicated, determine if the points form a parallelogram. If the opposite sides are
congruent, the slopes of opposite sides are congruent, or the diagonals share the same midpoint,
then the points form a parallelogram.
TOP: Prove that a set of points forms a parallelogram in the coordinate plane.
KEY: Parallelograms, Determining a Parallelogram
NOT: /A/ Correct! /B/ Which method was used to solve the problem?/C/ Which method was used
to solve the problem? /D/ This is not valid reason for the quadrilateral to be a parallelogram.
43. ANS: B
All sides of a rhombus are congruent.
TOP: Recognize and apply the properties of rhombi.
KEY: Rhombi, Properties of Rhombi
NOT: /A/ All sides of a rhombus are congruent. /B/ Correct! /C/ All sides of a rhombus are
congruent. /D/ Do not use properties of special triangles.
44. ANS: A
Plot the vertices on a coordinate plane. Determine if the diagonals are perpendicular. If so, the
quadrilateral is either a rhombus or square. Use the distance formula to compare the lengths of the
diagonals. If the diagonals are congruent and perpendicular, the quadrilateral is a square.
TOP: Recognize and apply the properties of squares.
KEY: Squares, Properties of Squares
NOT: /A/ Correct! /B/ Are the angles congruent? /C/ Remember to list all that apply./D/ Are the
sides congruent?
45. ANS: D
To find the other base, substitute the given values into the formula
.
TOP: Recognize and apply the properties of trapezoids.
KEY: Trapezoids, Properties of Trapezoids
NOT: /A/ Do not subtract the base from the median. /B/ AB is the median not a base. /C/ Do not
add the median and base./D/ Correct!
46. ANS: A
The circumference formula is Diameter  . The diameter shown also happens to be the diagonal
of a square, so it can be found by multiplying the side of the square by
. So the diameter would
be side 
and the circumference would be side   

TOP: Solve problems involving the circumference of a circle. KEY: Circles, Circumference
NOT: /A/ Correct! /B/ Check your diameter again. /C/ You need a radical 2 in your answer./D/
You need to double your radius.
47. ANS: B
is complementary with
, so its measure is 90 – 50 = 40.
TOP: Recognize major arcs¸ minor arcs¸ semicircles¸ and central angles and their measures.
KEY: Major Arcs, Minor Arcs, Semicircles, Central Angles
NOT: /A/ Did you subtract carefully? /B/ Correct! /C/ With what angle is it complementary? /D/
What is the measure of angle BAE?
48. ANS: B
Since
and
are congruent and perpendicular to separate chords, the chords
and
must
also be congruent. Additionally,
measure of
and
bisect these chords, so TS = RS. So take the given
and double it. That will be the measure of
and also
.
TOP: Recognize and use relationships between arcs and chords and diameters.
KEY: Arcs, Chords, Diameters
NOT: /A/ What is the relationship between PR and TS? /B/ Correct! /C/ Isn't PR longer than TS?
/D/ What is the relationship between PR and TS?
49. ANS: B
The measure of arc BC is 2 
. Since the full circle measures 360, then m arc AD is 360 –
(
+
+
). Finally, since
is an inscribed angle for arc
, its measure is half
m arc AD.
TOP: Find measures of inscribed angles. KEY: Inscribed Angles, Measure of Inscribed Angles
NOT: /A/ Can you find the measure of arc AD?/B/ Correct! /C/ Inscribed angles are half the
measure of the arc. /D/ Focus on arc AD.
50. ANS: B
The triangle shown is a right triangle since the tangent segment, FE, intersects a radius, DE, which
always results in a right angle. So to solve for x, use the Pythagorean Theorem. Note that
since they are both radii of the same circle. The equation from the Pythagorean Theorem is
.
TOP: Use properties of tangents.
KEY: Tangents
NOT: /A/ Use the Pythagorean Theorem and mDE = x./B/ Correct! /C/ Is the triangle a right? /D/
Use the Pythagorean Theorem and mDE = x.
51. ANS: C
When two secants intersect in the interior of a circle, then the measure of an angle formed by this
intersection is equal to one-half the sum of the measures of the arcs intercepted by the angle and its
vertical angle. In this diagram, the measure of one of the intercepted arcs for
is not given, but it
can be found since the sum of all of the arcs must be 360. Subtracting the sum of the other 3 arcs
from 360, leaves 110. That means
.
TOP: Find measures of angles formed by lines intersecting on or inside a circle.
KEY: Measure of Angles, Circles
NOT: /A/ Did you use the correct arcs? /B/ Add the intercepted arcs and divide by 2. /C/ Correct!
/D/ Did you find the measure of the unlabeled arc?
52. ANS: D
When two secants intersect in the exterior of a circle, then the measure of the angle formed is equal
to one-half the positive difference of the measures of the intercepted arcs. Here that gives the
equation
. Solving this equation results in the answer x = 10.
TOP: Find measures of angles formed by lines intersecting outside the circle.
KEY: Measure of Angles, Circles
NOT: /A/ How are the angle and the intercepted arcs related? /B/ Check your subtraction. /C/ Did
you subtract carefully? /D/ Correct!
53. ANS: C
When a secant and tangent intersect in the exterior of a circle, then the measure of the angle
formed is equal to one-half the positive difference of the measures of the intercepted arcs. Here
that gives the equation
. Solving this equation results in an answer of x = 25.
Note that the intercepted arc nearest the angle is 40 since 360 – 230 – 90 = 40.
TOP: Find measures of angles formed by lines intersecting outside the circle.
KEY: Measure of Angles, Circles
NOT: /A/ Check your subtraction. /B/ Did you subtract carefully? /C/ Correct. /D/ Check your
subtraction.
54. ANS: C
The products of the segments for each intersecting chord are equal, so 2  x = 3  6. Solving this
equation, x = 9.
TOP: Find measures of segments that intersect in the interior of a circle.
KEY: Circles, Interior of Circles
NOT: /A/ Use multiplication, not addition. /B/ Multiply the segments and set them equal to each
other. /C/ Correct! /D/ Multiply the segments and set them equal to each other.
55. ANS: D
When a secant segment and a tangent segment intersect in the exterior of a circle, set the product
of each external part of the secant segment and the entire secant segment equal to the square of the
tangent segment. Here that gives the equation x  x = 2  (2 + 6). Solving this equation results in
x = 4.
TOP: Find measures of segments that intersect in the exterior of a circle.
KEY: Circles, Exterior of Circles
NOT: /A/ Check your multiplication. /B/ You need to multiply, not add. /C/ Check the segments
in your multiplication. /D/ Correct!
56. ANS: A
When two secant segments intersect in the exterior of a circle, set an equality between the product
of each external segment and the entire segment. Here that gives the equation 2  12 = x  (x + 5).
Solving this equation results in x = –8, 3. Since x is a measurement and cannot be negative, the
answer is x = 3.
TOP: Find measures of segments that intersect in the exterior of a circle.
KEY: Circles, Exterior of Circles
NOT: /A/ Correct! /B/ Check your multiplication. /C/ You need to multiply, not add. /D/ Check
the segments in your multiplication.
57. ANS: A
The perimeter is the sum of all four sides. So add the two sides given and then double the result.
This accounts for the other two sides which are the same as those given. For example, if the two
given sides are 8 and 12, the perimeter is 2  (8 + 12) = 40. The area is base  height. The base is
given. The height can be found by recognizing the 30°-60°-90° triangle at the left. The height is
the long leg which is found by taking the hypotenuse (the given slanted side) and dividing by 2,
then multiplying by the square root of three. For example, if the slanted side is 8, the height is
82
. Multiply that height times the given base and you have the area.
TOP: Find areas of parallelograms.
KEY: Parallelograms, Area, Area of Parallelograms
NOT: /A/ Correct! /B/ Perimeter is the sum of all four sides. Area is base times height./C/
Perimeter is the sum of all four sides. /D/ Area is base times height.
58. ANS: A
The area of a rhombus is found by the formula
diagonals of the rhombus are 20 and 30, the area is
. For example, if the
 20  30 = 300.
TOP: Find areas of rhombi.
KEY: Area, Rhombi, Area of Rhombi
NOT: /A/ Correct! /B/ Area is diagonal 1 times diagonal 2 divided by 2./C/ How do you find the
area of a rhombus? /D/ What is the formula for the area of a rhombus?
59. ANS: D
This figure represents a rectangle attached to a right triangle. The area of the rectangle is
. The area of the triangle is
. So the total area of the figure is
.
TOP: Find areas of irregular figures.
KEY: Area, Irregular Figures, Area of Irregular Figures
NOT: /A/ The dimensions of the rectangle are 8 by 13. /B/ The area of the rectangle is 0.5 * 6 * 6.
/C/ The dimensions of the rectangle are 8 by 13. /D/ Correct!
60. ANS: D
This figure represents a rectangle with two identical semicircles cut out on each side. The
semicircles combine to make one circle with a diameter of 24 and a radius of 12. The area of the
rectangle is
. The area of the circle is
, using 3.14 for . So the area
of the figure is
.
TOP: Find areas of irregular figures.
KEY: Area, Irregular Figures, Area of Irregular Figures
NOT: /A/ The radius of the circle is 12, not 24. /B/ The two semicircles add up to one circle with
a radius of 12. /C/ The two semicircles add up to one circle with a radius of 12. /D/ Correct!
61. ANS: C
The formula for the lateral area of a regular pyramid is
where P is the perimeter of the base
and is the slant height of a face of the pyramid. For example, if the slant height is given as 12 and
the side as 14, then the perimeter is 56 (4 time 14, since it is a square), and the lateral area is
units2. The lateral area of the cube is perimeter times height. The
perimeter in this example as 56, and the height is 14 since it is a cube. So the lateral area of the
cube is
. Finally, the total lateral area of the solid is the sum of the lateral area of the
pyramid and the lateral area of the cube. Add the two previous answers to get
units2.
TOP: Find lateral areas of regular pyramids.
KEY: Lateral Area, Pyramids, Lateral Area of Pyramids
NOT: /A/ The lateral area is the sum of the lateral areas of the pyramid and the cube. /B/ You
found the surface area, not the lateral area. /C/ Correct! /D/ The lateral area is the sum of the lateral
areas of the pyramid and the cube.
62. ANS: A
The formula for the surface area of a regular pyramid is
+ B where P is the perimeter of the
base, is the slant height and B is the area of the base of the pyramid. Since this is a regular (or
equilateral) triangle, to find the perimeter of the base, multiply the length of one side by 3. To find
the area of the triangular base, the formula is
. To find the length of the slant height, use
the Pythagorean Theorem as follows:
. For example, if the lateral
edge is given as 16 and the side as 18, then the slant height is
surface area is
. The
units2.
TOP: Find surface area of regular pyramids.
KEY: Surface Area, Pyramids, Surface Area of Pyramids
NOT: /A/ Correct! /B/ What is the formula for finding the surface area of a regular pyramid? /C/
The surface area of a regular pyramid is one-half of the perimeter of the base times the slant height
+ the area of the base./D/ You only found the lateral area. You need the surface area.
63. ANS: B
The formula for the lateral area of a cone is r where r is the radius of the circular base and is
the slant height of the cone. For example, if the slant height is given as 10 and the radius as 6, then
the lateral area is then computed as
units2.
TOP: Find lateral areas of cones.
KEY: Lateral Area, Cones, Lateral Area of Cones
NOT: /A/ What is the formula for the lateral area of a cone? /B/ Correct! /C/ You found the
surface area. You need to find the lateral area. /D/ The lateral area of a cone is pi times radius
times slant height.
64. ANS: C
A plane that intersects a sphere through its center will result in a great circle by definition.
TOP: Recognize and define basic properties of spheres.
KEY: Spheres
NOT: /A/ The intersection will result in a great circle. /B/ The intersection will result in a great
circle. /C/ Correct! /D/ The intersection will result in a great circle.
65. ANS: A
The radii of a sphere, like those of a circle, are all the same length. So the statement is true.
TOP: Recognize and define basic properties of spheres.
KEY: Spheres
NOT: /A/ Correct! /B/ All radii of the same sphere are the same length. /C/ All radii of the same
sphere are the same length. /D/ This is not a relevant counterexample.
66. ANS: A
The surface area of a sphere is found by the formula
, where r is the radius of the sphere. This
diagram gives the diameter, so begin by computing the radius which is half the diameter. For
example, if the diameter is given as 12, then the radius is 6 and the surface area would be
.
TOP: Find surface area of spheres.
KEY: Surface Area, Spheres, Surface Area of
Spheres
NOT: /A/ Correct! /B/ Does the formula indicate cubing the radius?/C/ Did you use the correct
formula? /D/ Did you leave pi out of the formula?
67. ANS: D
The surface area of a sphere is found by the formula
, where r is the radius of the sphere. This
problem gives the circumference, so begin by computing the radius which can be found by the
formula
. For example, if the circumference is given as 12.56, then the radius is
the surface area would be
surface area is
and
. If the solid is a hemisphere, then the formula for the
, since a hemisphere has half the surface area of the sphere
plus the area of the great circle on its base
.
TOP: Find surface area of spheres.
KEY: Surface Area, Spheres, Surface Area of
Spheres
NOT: /A/ Does the formula indicate cubing the radius?/B/ Did you leave pi out of the formula?
/C/ Did you use the correct formula? /D/ Correct!
68. ANS: C
The volume of the cone is
cm3. The volume of the ice (a sphere) is
cm3. So the volume of the ice is greater than the volume of the cone, causing
it to overflow the cone.
TOP: Solve problems involving volumes of spheres.
KEY: Volume, Spheres, Volume of Spheres
NOT: /A/ Check your volume calculations. /B/ Check your volume calculations. /C/ Correct! /D/
This problem can be solved with the information given.
69. ANS: A
The formula for the volume of a sphere is
. So for this sphere, the volume is
cm3.
TOP: Solve problems involving volumes of spheres.
KEY: Volume, Spheres, Volume of Spheres
NOT: /A/ Correct! /B/ The formula has a 4/3, not a 1/3. /C/ You have to cube the radius. /D/ The
formula has a 4/3, not a 1/3. You also need to cube the radius.
70. ANS: B
Two spheres that are congruent are the same size and shape. Therefore, they would have the same
surface area. So this statement is always true.
TOP: State the properties of similar solids.
KEY: Similar Solids
NOT: /A/ Congruent means same size and shape. /B/ Correct! /C/ Congruent means same size
and shape.