Geometry B Exam Review 2015-2016

GEOMETRY B
Semester Exam Review
Geometry B
Exam Review
2015-2016
Notes to the student: This review prepares you for the semester B Geometry Exam. The exam
will cover units 3, 4, and 5 of the Geometry curriculum.
The exam consists of two parts. Part 1 is 15 selected response items each worth 2 points. The
second part is 15 extended response items worth varying numbers of points for a total of 70
points.
A calculator may be used on both parts of the exam.
Many answers will be in terms of  or in radical form. You may leave answers in that form or
approximate as a decimal to two places past the decimal point. For most accurate
approximations use the  key on your calculator. Do not use 3.14 for  . On the next two pages
you will find the formulas that will be available to you in the exam booklet.
FIGURES ARE NOT DRAWN TO SCALE!!!!!
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Page 1
GEOMETRY B
Semester Exam Review
Area/Circumference
1
Triangle: A  bh
2
Rectangle: A  bh
Parallelogram: A  bh
1
Regular Polygon: A   apothem  perimeter
2
Circle Area: A   r 2
Circle Circumference: C  2 r   d
Trapezoid: A 
1
 b1  b2  h
2
Volume
Prism/Cylinder:
V  Bh  area of base  height
Pyramid/Cone:
1
1
V  Bh   area of base  height
3
3
Sphere:
4
V   r3
3
Density Formula
Density = Mass  Volume
Coordinate Geometry
Slope:
y2  y1
x2  x1
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 x  x y  y2 
Midpoint:  1 2 , 1

2 
 2
Distance:
 x2  x1    y2  y1 
2
2
Page 2
GEOMETRY B
Semester Exam Review
Conic Sections
Equation of a circle with center at  h, k  and radius r:
 x  h   y  k 
2
2
 r2
Equations of a parabola with vertex at the origin, with p the distance from the vertex to the focus
and vertex to directrix:
x 2  4 py or y 
1 2
x ;
4p
opens up if p  0 , opens down if p  0
y 2  4 px or x 
1 2
y ;
4p
opens right if p  0 , opens left if p  0
Circles
Arc Length (degrees): S 
x
 2 r 
360
Arc Length (radians): S  r
Sector Area (degrees): A 
x
 r2 

360
Sector Area (radians): A 
 

 r2
2
Angle and Arc Formulas
B N
A L D
H
G
C mA 
1 radian =
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180

1 
mBC
2
degrees
J K F
E
M mGDE 

1 

mGE  mFH
2
1 degree =

180

mJ 

1 

mMN  mLK
2

radians
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GEOMETRY B
Semester Exam Review
Unit 3
1.
Which pairs of figures have the same volume?
a.
b.
5
r  4 5
5
5
Bases are squares
For items 2 and 3, find the volume if each figure is revolved about the dashed line.
2.
3.
3
4
5 4.
7
Name the cross section that is formed in each case.
a.
A cone, cut by a plane parallel to the base, not through the apex (top).
b.
A cone. cut by a plane perpendicular to the base, through the apex (top).
c.
A rectangular pyramid, cut by a plane parallel to the base, not through the apex
(top).
d.
A square pyramid, cut by a plane parallel to the base, not through the apex (top).
e.
A cylinder, cut by a plane parallel to the base.
f.
A cylinder, cut by a plane perpendicular to the base.
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GEOMETRY B
Semester Exam Review
5.
What is the relationship between the volumes of a cone and a cylinder if the cone and
cylinder have the same radii and heights?
6.
What is the relationship between the volumes of a pyramid and a prism, if the pyramid
and prism have the same base areas and heights?
7.
A scoop of ice cream is in the shape of a sphere with radius 3 cm is placed in a cone that
has a radius of 2 cm and a height of 9 cm. If the ice cream melts into the cone, will the
ice cream overflow the cone? Show how you determined your answer.
8.
The scoop of ice cream in item 8 has a mass of 124 grams. What is the density (in
g/cm3 ) of the ice cream?
9.
A movie theater sells popcorn in pyramid-shaped boxes whose base is a square of side 15
centimeters and a height of 25 centimeters.
a.
What is the volume of the popcorn box?
b.
The density of popcorn is 0.03 grams per cubic centimeter. What is the mass of
the popcorn in the box?
c.
When popcorn is popped, it is stored in a cube that is 45 centimeters on each
side. How many boxes of popcorn can be filled from this container?
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GEOMETRY B
10.
Semester Exam Review
The three figures below have the same volume. Find the missing height in each case.
a.
b.
h
6
12
3
h
2
3
3
h = ______
24
h = ________
For items 11 through 13, find the volume of each solid described.
11.
A sphere with radius 6 cm.
12.
A cone with radius 5 cm and height 9 cm.
13.
A square pyramid with base of side 8 cm and height 3 cm.
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GEOMETRY B
14.
Semester Exam Review
A company is producing a special part for a machine. The part consists of a cylinder of
tin (white) that is inside of another cylinder made of copper (shaded). The part is shown
below.
6 mm
Copper
5 mm
4 mm
3 mm
Tin
a.
What is the total volume of the entire part? You may give your answer in terms
of  or to the nearest cubic millimeter.
b.
What is the volume of the tin used in the part? You may give your answer in
terms of  or to the nearest cubic millimeter.
c.
What is the volume of copper used in the part? You may give your answer in
terms of  or to the nearest cubic millimeter.
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GEOMETRY B
Semester Exam Review
Unit 4
Use the word bank to complete the following definitions. Words may be used more than once.
Word bank:
center focus directrix radius equidistant
15.
A circle is the set of points in the plane that are _________________ from a given point,
called the _______________. The distance from the given point to every point on the
circle is called the ____________.
16.
A parabola is a set of points in the plane that are ___________ from a given point,
called the _________, and a given line, called the ____________.
For items 17 and 18, sketch graphs of the following circles.
17.
x2  y2  4
18.
 x  2
2
y
O © MCPS
  y  1  9
2
y
x
O
x
Page 8
GEOMETRY B
Semester Exam Review
For 19 and 20, write equations for the following circles and state the circumference and area of
each circle.
19.
20.
y
y
O 21.
O
x
x
Equation: _________________________
Equation: _______________________
Circumference: _____________
Circumference: ____________
Area : _______________
Area: _____________
2
2
Is the point  5,3 on the circle whose equation is  x  3   y  2   65 ? Show how
you determined your answer.
22.
For the equation of a circle x 2  y 2  6 y  27 , complete the square to put it in the form
 x  h 2   y  k 2  r 2
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, then give the center and the radius.
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GEOMETRY B
Semester Exam Review
For items 23 through 25, sketch graphs of the following parabolas. For each graph, also graph
the focus and directrix.
23.
x2  6 y
24.
25. x 2  4 y
y 2  8x
y
y
O y
O
x
O x
x
For items 26 and 27, write the equation for the parabola.
26.
27.
y
y
F  0, 2  O © MCPS
F  2.5, 0  x
O
x
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GEOMETRY B
28.
Semester Exam Review
Look at the parabola below.
y 10
9
8
7
6
5
4
F 3
2
1
O
‐10 ‐ 9 ‐ 8 ‐ 7 ‐ 6 ‐ 5 ‐ 4 ‐ 3 ‐ 2 ‐ 1 1 2 3 4 5 6 7 8 9 10
‐1
x ‐2
‐3
‐4
‐5
‐6
‐7
‐8
‐9
‐10
Show that the point P  8,8  satisfies the definition of the parabola by showing that the
distance between the focus and the point P  8,8  is equal to the distance from the point
P  8,8  to the directrix.
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GEOMETRY B
29.
Semester Exam Review
How many points of intersection do the graphs of the circle x 2  y 2  16 and the
parabola y  x 2 have? You may use the grid below to help you.
y
O
30.
x
2
How many points of intersection do the graphs of the circles  x  2   y 2  4 and
 x  2 2  y 2  4
have? You may use the grid below to help you.
y
O
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x
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GEOMETRY B
Semester Exam Review

Look at the graph of line AB on the coordinate plane below.
y 10
9
8
7
6
B 10,8  5
4
3
2
1
O
‐10 ‐ 9 ‐ 8 ‐ 7 ‐ 6 ‐ 5 ‐ 4 ‐ 3 ‐ 2 ‐ 1 1 2 3 4 5 6 7 8 9 10 ‐1
x ‐2
‐3
‐4
A  10, 7  ‐5
‐6
‐7
‐8
‐9
‐10
31.
What is the length of AB ?
32.
Point C is between points A and B and has coordinates  2, 2  . What is the ratio AC : CB ?
33.
Determine the coordinates of point D so that the ratio AD : DB  1: 4 .
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GEOMETRY B
34.
Semester Exam Review
Look at line k whose equation is y  3x  2 on the coordinate plane below.
k
y
O
x
Write equations for the following lines.
a.
Parallel to line k, passing through the point  0, 2  .
b.
Perpendicular to line k, passing through  0, 5 .
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GEOMETRY B
Semester Exam Review
Items 35 through 38 (this item continues on the next page) uses the quadrilateral below.
y
A 1, 4  B  5,1 D  2,0  O
x
C  2, 3 Kelly thinks that the figure might be a square. Remembering her geometry she does the
following.
35.
Kelly first needs to know if the figure is a parallelogram. Show that the figure is a
parallelogram by showing that the slopes of opposite sides are equal.
36.
Kelly now wants to determine if the figure is a rectangle. There are two ways to do this.
a.
Show, by using slopes, that one of the angles is a right angle, and therefore there
are four right angles and the figure is a rectangle.
b.
Show that the diagonals of the parallelogram are congruent, and the figure is a
rectangle.
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GEOMETRY B
37.
38.
Semester Exam Review
Now that Kelly knows the figure is a rectangle. If she can show that it is a rhombus then
it must be a square. Again there are two ways to do this.
a.
Show that all four sides are the same length, and therefore the figure is a rhombus.
b.
Show, by using slope computations that the diagonals are perpendicular, and
therefore the figure is a rhombus.
Kelly also remembers a property of parallelograms that the diagonals bisect each other.
Show that this true by determining the midpoint of each diagonal and showing that each
midpoint has the same coordinates.
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GEOMETRY B
Semester Exam Review
For items 39 and 40, determine the area and perimeter of each shaded figure below. Each grid
line represents 10 meters.
39.
40.
y y
80
80
60
50
20
10
20
10
0 10 20 30 40 50 60 70 80 90 100 x
0 10 20 30 40 50 60 70 80 90 100 x
Area: ____________________
Area: _____________________
Perimeter: ________________
Perimeter: _________________
Unit 5
For items 41 through 49, find the value of x in each figure below.
41.
43. C is the center
42.
x
o
50o
x
160o
o
C
xo
70o
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GEOMETRY B
Semester Exam Review
44.
46. AB is a diameter
45.
o
60
xo
o
28
xo
60o
B
85o
A
xo
47.
48.
49.
290o
xo
50o
x
o
20o
40o
70o
35o
xo
50.
Quadrilateral ABCD is inscribed in the circle below.
A
B
D
C
a. mA  mC  ________ . Explain your reasoning.
b. If AC is a diameter, what is mD or mB ? Explain your reasoning.
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GEOMETRY B
Semester Exam Review
Look at the circle O below.
A
140o
20o
60o
O
B
F
E
42o
C
Determine the following:
D
51.
m
AB _______
52.
 _______
mEF
53.
 _______
mBC
54.
mAFB _______
55.
mFDB ______
56.
mFEA _______
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GEOMETRY B
Semester Exam Review
Look at circle P below. Line m is tangent to the circle at point R.
PR  8, PS  10, UW  TZ
T
m
S
X
W
U
R
P
Z
Complete the following.
57.
mPRS  ________
59.
TU is congruent to which segment? ____________
60.
WTU is congruent to which angle? _________________
61.
 is congruent to which arc? __________
TW
58.
SR  ________
62.
SX  _________
63.
What part of the circumference of a circle is represented by arc whose measure is 
radians?
64.
What part of the circumference of a circle is represented by an arc whose measure is

3
radians?
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GEOMETRY B
Semester Exam Review
In items 65 and 66 below, find the radian measure of the central angle and the area of the sector.
65.
66.
24 cm
F
6 cm
C
P
O
D
8 cm
3 cm
Q
Radian measure of COD ________
Radian measure of FPQ ____________
Area of sector COD ______________
Area of sector FPQ ________________
In items 67 and 68 below, find the length of arc and area of sector.
67.
68.
S
R
120o
o
T
40
9 cm
W
X
3 cm
Z
 _______________
Length of SW
 _____________
Length of RZ
Area of sector STW _______________
Area of sector RXZ _____________
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