Geometry (2nd semester)
Review for Final Exam
Name: ____________________________
Problems 1-4: Solve for x. Give simplified, exact answers.
1.
5 – (x – 3) = x + 12
2.
2x + 3(x – 2) = 4x – 5
3.
5x2 + 6x = 8
4.
x(x – 2) = 2
5. Find the degree measure of angle ACB:
B
3x + 15
2x
4x + 34
A
6.
C
C
Find BC:
2x – 1
A
7.
x+1
B
2x
D
If segment PQ is parallel to segment BC, find the length of segment AB:
A
2x + 1
x+2
Q
P
x+2
x
B
C
Page 1 of 12
8.
Find DE:
D
B
2x + 3
3x
A
4x – 1
C
5x + 2
5x
E
9.
If segment MN is the midsegment of trapezoid
2x
A
B
ABCD, find the length of segment CD:
2x + 3
M
x2 – 9
D
10.
N
C
A wire is bent to form a rectangle. If the length of this rectangle is 5 inches less than twice its
width and the area of the rectangle is 42 square inches, then how long, in inches, was the wire
before it was bent?
Problems 11-24: Find the exact value of x. Simplify the answer.
11.
12.
x
2 15
30
o
60
9
x
13.
45
x
14.
o
x
3 2
o
6
45
Page 2 of 12
o
H
15.
G
16.
B
A
3x
3x +
6
x
1
E
F
6
x
2 3
D
x–1
C
5
17.
5
18.
2
5x
x
O
x
x
4x + 3
19.
3
3
20.
4
x
2
x
6
24
24o
21.
22.
xo
x
8
110o
12
Page 3 of 12
160o
23.
160o
24.
30o
xo
30o
xo
25.
Refer to the circle at the right.
B
A
a. Find the degree measure of AC.
75
o
115
O
o
b. Find the degree measure of ABC.
C
c. If the circumference of
26.
27.
O is 72 cm, then what is the length of BC?
In an arcade game, the “monster” is the shaded sector of a
circle of radius 1 cm, as shown in the figure. The missing
piece (the mouth) has a central angle of 60o. What is the
area of the monster in cm2?
m
1c
o
60
The diameter of circle O is 6 units. Segment AB is 1 unit
P
long, and segment BP is perpendicular to segment AB.
How long is segment BP to the nearest thousandth of a
unit?
Page 4 of 12
A
B
O
28.
If you reflect a geometric figure in two intersecting lines that make an angle of 30o with each
other, then the result is the same as:
a. a rotation through 30o
b. a rotation through 60o
c. a rotation through 15o
d. a rotation through 90o
29.
What is the result of reflecting a geometric figure in two parallel lines that are 3 cm apart?
30.
The regular pentagon PQRST is rotated counterclockwise
Q
so that vertex P is mapped onto vertex S. Through what
P
angle was the pentagon rotated?
R
S
T
Problems 31-32: Each figure shows a segment and its image under a dilation. Give the scale factor
of each dilation.
31.
8 cm
Preimage
32.
Center of
dilation
9 in
Center of
dilation
6 cm
1 ft
Preimage
Image
Image
33.
What is the distance between the points (3, –5) and (–2, 7)?
34.
Given: A(2, 1), B(6, 4), C(8, 0)
Which kind of triangle is ABC , an acute triangle, an obtuse triangle, or a right triangle?
Page 5 of 12
35.
What are the coordinates of the centroid of PQR if P(0, 0), Q(24, 0), and R(18, 12)?
36.
What is the degree measure of angle D?
A
75
o
110
D
37.
o
B
C
Q
If quadrilateral PQRS is cyclic, what is the measure of P ?
92
o
P
63
o
S
38.
39.
Which of the following intersect at the center of a circle inscribed in a triangle?
a. the medians
b. the altitudes
c. the angle bisectors
d. the perpendicular bisectors of the sides
Which of the following intersect at the center of a circle circumscribed about a triangle?
a. the medians
b. the altitudes
c. the angle bisectors
d. the perpendicular bisectors of the sides
Page 6 of 12
R
40.
Triangle ABC is equilateral, AP = 10, and PC = 6.
How long is BP?
A
10
B
C
6
P
41.
What is the locus of points that are equidistant from two given points, A and B?
42.
What is the locus of points that are equidistant from the sides of an angle?
43.
The bases of a right prism are isosceles right triangles.
The prism is 15 cm long, and the area of each base
is 36 cm2.
a. What is the volume of that prism?
15 cm
b. What is the total surface area of that prism?
Page 7 of 12
area of base = 36 cm2
44. An oblique rectangular prism has square ends on the
left and right, slanting 60o with the horizontal. Its
6 cm
rectangular base measures 15 cm by 6 cm. The
60
picture below is a net for this prism:
o
15 cm
15 cm
6 cm
6 cm
6 cm
60
Top
Back
o
6 cm
15 cm
6 cm
Left
Right
Bottom
60
6 cm
o
Front
15 cm
45.
a.
What is the total surface area of this prism?
b.
What is the volume of this prism?
Refer to the object on the right, which is made
of six cubes glued together. Each of these six
cubes is 1 cubic inch.
a.
Draw the front view. Be sure to show any
Front
hidden edges with dashed lines.
b.
What is the total surface area of this object?
c.
What is the volume of this object?
Page 8 of 12
6 cm
6 cm
46.
A cone and a cylinder have equal base
areas and heights, and the height and
base-diameter of each object is 12 cm.
a. What is the ratio of the cone’s volume
to the cylinder’s volume?
b. What is the lateral surface area of the cylinder?
c. What is the lateral surface area of the cone?
47. A cross-section of a sphere (when cut by a plane through its center) is a circle of area 100 cm2.
a. What is the surface area of the sphere?
Circle:
2
Area = 100 cm
b. What is the radius of the sphere?
c. What is the volume of the sphere?
48. The base of a right square pyramid has an area of
100 square inches, and its altitude is 12 inches.
12 in
a. What is the volume of this pyramid?
b. What is the slant height of this pyramid?
c. What is the lateral surface area of this pyramid?
d. What is the total surface area of this pyramid?
Page 9 of 12
49.
In the figure on the right, the small cone is formed by
4 in
cutting off the lower part of the larger cone. If the
larger cone (whole cone) has a weighs 405 ounces,
8 in
what is the weight of the smaller cone? Give the
number of ounces.
50.
An open box is made of ¾ inch thick wood. The base is
square with outside measurements of 1 ft on each edge, and
the height measures 10 inches on the outside.
a.
10 in
Using the outside measurements, what is the volume
of the box?
1 ft
51.
b.
What is the capacity of the box (volume of the inside)?
c.
What is the volume of wood used to make the box?
1 ft
B
Find tan A in the triangle on the right:
8 in
C
52.
1 ft
Q
3
If tan P  , then how long is PQ?
4
P
12
Page 10 of 12
R
A
53.
If the hypotenuse of a right triangle is three times as long as the shorter leg, then what is the
measure of the smallest angle in this triangle?
54.
If one of the acute angles of a right triangle is 3 times as large as the other acute angle, and the
shorter leg is 12 cm, then how long is the hypotenuse, to the nearest hundredth of a cm?
18
55.
Find θ to the nearest tenth of a degree:
11
θ
23
56.
Find x to the nearest hundredth of a unit:
5
35
o
x
57.
Find x in this triangle. Round your answer to the
61
10
nearest one-hundredth of a unit.
22
Page 11 of 12
o
o
x
58.
What is the measure of angle A to the nearest degree?
A
8.5
3.6
67
59.
o
Find x to the nearest hundredth of an inch:
27
12 in
o
5 in
x
2
?
3
60.
What is the exact value of cos
61.
What is the area of a sector of 72o in a circle with
radius 20 cm?
o
72
20 cm
62.
What is the area of the triangle on the right?
9 cm
63.
107
o
Find the area of the triangle whose sides measure 15 cm, 17 cm and 21 cm.
Page 12 of 12
6 cm