GEOMETRY FINAL EXAM STUDY PROBLEMS
TRANSFORMATIONS
12-91. Multiple Choice: Assume that A(6, 2), B(3, 4), and C(4, −1) . If ΔABC is
rotated 90° counterclockwise (
) to form ΔA′B′C′ , and if ΔA′B′C′ is reflected
across the x-axis to form ΔA″B″C″ , then this is the coordinates of C″
1.
2.
3.
4.
(1, 4)
(−4, 1)
(1, −4)
(4, 1)
ANGLE RELATIONS
12-26. For each relationship below, write and solve an equation for x. Justify
your method.
1.
2.
12-46. Multiple Choice: In the diagram at right, the
value of x is:
1.
2.
3.
4.
1
2
3
4
None of these
3.
GEOMETRY FINAL EXAM STUDY PROBLEMS
12-71. For each situation below, decide if a is greater, b is greater, if they are the
same value, or if not enough information is given.
1. a is the measure of a central angle of an equilateral triangle; b is the
measure of an interior angle of a regular pentagon.
2.
3.
4. a = b + 3
5.
TRIANGLES
TRIANGLE PROPERTIES
12-10. Examine the diagram at right.
a) Write an equation using the geometric relationships in the
diagram. Then solve your equation for x.
b) Find the measures of the acute angles of the triangle. What
tool(s) did you use?
2
GEOMETRY FINAL EXAM STUDY PROBLEMS
TRIANGLE SIMILARITY
12-103. Examine the triangles below. Which, if any, are similar? Which are
congruent? For each pair that must be similar, state how you know. Remember
that the diagrams are not drawn to scale.
1.
2.
3.
4.
3
GEOMETRY FINAL EXAM STUDY PROBLEMS
TRIANGLE CONGRUENCIES
12-12. Multiple Choice: Based on the markings in the diagrams below, which
statement is true?
1.
2.
3.
4.
5.
ΔABC ≅ ΔXYZ
ΔABC ≅ ΔYXZ
ΔABC ≅ ΔZXY
ΔABC ≅ ΔZYX
None of these
MIDSEGMENTS OF TRIANGLES
12-53. In the triangle at right,
ΔABC and
is a midsegment of
is a midsegment of ΔADE.
1. If DE = 7 cm, find BC and FG.
2. If the area of ΔAFG is 3 cm2, what is the area of DECB?
RIGHT TRIANGLES
12-63. Examine the triangles below. Decide if each one is a right triangle. If the
triangle is a right triangle, justify your conclusion. Assume that the diagrams are
not drawn to scale.
TRIANGLE INEQUALITY
12-67. Multiple Choice: Which number below could be the length of the third
side of a triangle with sides of length 29 and 51?
1.
2.
3.
4.
4
10
18
23
81
GEOMETRY FINAL EXAM STUDY PROBLEMS
TRIGONOMETRY
12-30. Examine the diagram at right.
1. Explain why y = sinθ and x = cosθ.
2. According to this diagram, what is (sinθ)2 + (cosθ)2? Explain
how you know.
3. Does this relationship appear true for all angles? Use your
calculator to find (sin23º)2 + (cos23º)2 and (sin81º)2 + (cos81º)2. Write down
your findings.
11-19. Examine the diagram of the triangle at right.
1. Write an equation representing the relationship between x,
y, and r.
2. Write an expression for sinθ. What is sinθ if r = 1?
3. Write an expression for cosθ. What is cosθ if r = 1?
RATIOS OF SIMILARITY
12-72. In the diagram at right, ABCD ~
DCFE. Solve for x and y. Show all work.
PROOFS
CL 12-114. Use the diagram at right to prove
the following statement. Use the proof format,
two-column or flowchart, that you prefer.
5
 If
≅
then
⊥
and
.
≅
,
GEOMETRY FINAL EXAM STUDY PROBLEMS
POLYGONS
12-34. Multiple Choice: What is the measure of each interior angle of a regular
octagon?
1.
2.
3.
4.
135°
120°
180°
1080°
12-89. If the sum of the interior angles of a regular polygon is 2160°, how many
sides must it have?
12-101. Find the area of each quadrilateral below. Show all work.
1. Kite
2. Rhombus
12-61. In ΔPQR at right, what is m∠Q? Explain how you found your answer.
6
GEOMETRY FINAL EXAM STUDY PROBLEMS
12-86. Find the area of the shaded region of the regular
pentagon at right. Show all work.
12-62. The United States Department of Defense is
located in a building called the Pentagon because it is in the shape of a regular
pentagon. Known as “the largest office building in the world,” its exterior edges
measure 921 feet. Find the area of land enclosed by the outer walls of the
Pentagon building.
12-92. Find the area and perimeter of the shape at
right. Assume that any non-straight portions of the
shape are part of a circle. Show all work.
CIRCLES
12-11. Use the diagram of
questions below.
at right to answer the
1. If m∠x = 28º, what is
2. If AD = 5 and BD = 5
?
, what is the area of
?
3. If the radius of
what is BD?
is 8 and if
= 100º,
12-77. Multiple Choice: The radius of the front wheel of Gavin’s tricycle is 8
inches. If Gavin rode his tricycle for 1 mile in a parade, approximately how many
rotations did his front wheel make? (Note: 1 mile = 5280 feet).
1.
2.
3.
4.
7
50
1260
660
42,240
GEOMETRY FINAL EXAM STUDY PROBLEMS
11-20. In a circle, chord
has length 10 units, while
of the circle? Draw a diagram and show all work.
11-44. Find the area and circumference of
right. Show all work.
= 60°. What is the area
at
12-33. Use all your circle relationships to solve for the variables in each of the
diagrams below. Assume that C is the center of the circle for parts (b) and (c).
1.
and
2. The area of
intersect at E.
is 25π sq. units
INSCRIBED ANGLES
12-22. The figure at right is a pentagram. A pentagram is a
5-pointed star that has congruent angles at each of its outer
vertices.
1. Use the fact that all pentagrams can be inscribed in a
circle to find the measure of angle a at right.
2. Find the measure of angles b, c, and d.
8
GEOMETRY FINAL EXAM STUDY PROBLEMS
TANGENTS AND SECANTS: ARCS AND ANGLES
12-52. Use the relationships in each diagram below to solve for the given variables.
1.
2. The area of
3. The diameter of
is 36π sq. units.
is 13 units. w is the length of
4. POLYGON PROPERTIES
9
.
GEOMETRY FINAL EXAM STUDY PROBLEMS
CL 12-115. Examine the diagrams below. For each one, use geometric
relationships to solve for desired information.
1. 
2.
3.
4.
3D SOLIDS
PYRAMIDS
12-107. Multiple Choice: A square based pyramid has a slant height of 10 units
and a base edge of 10 units. What is the height of the pyramid?
1. 5
2. 5
3. 6
4. 8
10
GEOMETRY FINAL EXAM STUDY PROBLEMS
VOLUME
12-7. Find the volume of each shape below. Assume that all corners in part (b) are
right angles.
1. Cone
2. Complex solid
12-21. A silo (a structure designed to store grain) is designed as a
cylinder with a cone on top, as shown in the diagram at right.
1. If a farmer wants to paint the silo, how much surface area
must be painted?
2. What is the volume of the silo? That is, how many cubic
meters of grain can the silo hold?
SURFACE AREA
12-28. Find the surface area of the solids below. Assume that the solid in part (a) is a
prism with a regular octagonal base and the pyramid in part (b) is a square-based
pyramid. Show all work.
1.
2.
11
GEOMETRY FINAL EXAM STUDY PROBLEMS
12-40. Cawker City, Kansas, claims to have the world’s
largest ball of twine. Started in 1953 by Frank Stoeber,
this ball has been created by wrapping more than 1300
miles of twine. In fact, this giant ball has a circumference
of 40 feet. Assuming the ball of twine is a sphere, find
the surface area and volume of the ball of twine.
11-10. The lateral surface of a cylinder is the surface connecting the bases. For
example, the label from a soup can would represent the lateral surface of a
cylindrical can. If the radius of a cylinder is 4 cm and the height is 15 cm, find
the lateral surface area of the cylinder. Note: It may help you to think of
“unrolling” a soup can label and finding the area of the label.
12-54. Find the volume and lateral surface area of a cone if the circumference of
the base is 28π inches and the height is 18 inches.
CL 12-112. When considering new plans for a covered baseball stadium,
Smallville looked into a design that used a cylinder with a dome in the shape of a
hemisphere. The radius of the proposed cylinder is 200 feet and the height is 150
feet. See a diagram of this below.
1. One of the concerns for the citizens of Smallville is the cost of heating the
space inside the stadium for the fans. What is the volume of this
stadium? Show all work.
2. The citizens of Smallville are also interested in having the outside of the
new stadium painted in green. What is the surface area of the
stadium? Do not include the base of the cylinder.
12
GEOMETRY FINAL EXAM STUDY PROBLEMS
CL 12-113. An ice-cream cone is filled with ice cream. It also has ice-cream on
top that is in the shape of a cylinder. It turns out that the volume of ice cream
inside the cone equals the volume of the scoop on top. If the height of the cone is
6 inches and the radius of the scoop of ice cream is 1.5 inches, find the height of
the extra scoop on top. Ignore the thickness of the cone.
SIMILAR SOLIDS
12-50. A solid with volume 820 cm3 is reduced proportionally with a linear scale
factor of
13
. What is the volume of the resulting solid?