1
ALGEBRA CONCEPTS PA CORE 8 – COURSE 3
STUDENT WORKBOOK
UNIT 4 – GEOMETRY
STUDY
ISLAND
TOPICS
Name: _______________________________ Period ____
Unit 4
Geometry
PURPLE
GREEN
RED
5.5
The Pythagorean Theorem
11.2
11.7
5.6
Use the Pythagorean Theorem
11.3
11.8
5.7
Distance on the Coordinate Plane
11.3
11.7
6.1
Translations
9.8
6.2
Reflections
9.9
6.3
Rotations
9.1
6.4
Dilations
7.1
Congruence and Transformations
9.5
7.2
Congruence
9.5
7.3
Similarity and Transformations
7.4
Properties of Similar Polygons
7.5
Similar Triangles and Indirect
Measurement
A-1
7.6
Slope and Similar Triangles
A-1
8.1
Volume of Cylinders
10.7
8.2
Volume of Cones
10.9
8.3
Volume of Spheres
10.9
Object Transformation
Similarity and Congruence
Pythagorean Theorem
Volume
1
PRE ALGEBRA 2 -
Lesson 1 Skills Practice
Translations
Graph the image of the figure after the indicated translation.
1. 2 units left and 2. 4 units right and 3. 1 unit left and 4. 5 units right and
3 units up 1 unit up 2 units down 3 units down
Graph the figure with the given vertices. Then graph the image of the figure after the indicated translation and
write the coordinates of its vertices.
2
3
Lesson 1 Problem-Solving Practice
Translations
1. BUILDINGS The figure shows an outline of the White
House in Washington, D.C., plotted on a coordinate system.
Find the coordinates of points C and D after the figure is
translated 2 units right and 3 units up.
2. BUILDINGS Refer to the figure in Exercise 1. Find the
coordinates of points C and D after the figure is translated 1
unit left and 4 units up.
3. ALPHABET The figure shows a capital “N” plotted on a
coordinate system. Find the coordinates of points F and G
after the figure is translated 2 units right and 2 units down.
4. ALPHABET Refer to the figure in Exercise 3. Find the
coordinates of points F and G after the figure is translated 5
units right and 6 units down.
5. QUILT The beginning of a quilt is shown below. Look for a
pattern in the quilt. Copy and translate the quilt square to
finish the quilt.
6. BEACH Tylia is walking on the beach. Copy and translate
her footprints to show her path in the sand.
3
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Translations of Shapes
Graph the image of the figure using the transformation given.
1) translation: 1 unit left
x
y
Q
X
G
U
2) translation: 1 unit right and 2 units down
x
y
I
T
E
3) translation: 3 units right
x
y
M
Y
T
Q
4) translation: 1 unit right and 2 units down
x
y
G
W
E
5) translation: 5 units up
U
(
−3, −4
)
, M
(
−1, −1
)
, L
(
−2, −5
)
x
y
6) translation: 3 units up
R
(
−4, −3
)
, D
(
−4, 0
)
, L
(
0, 0
)
, F
(
0, −3
)
x
y
-1-
4
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D
Worksheet by Kuta Software LLC
Find the coordinates of the vertices of each figure after the given transformation.
7) translation: 2 units left and 1 unit down
Q
(
0, −1
)
, D
(
−2, 2
)
, V
(
2, 4
)
, J
(
3, 0
)
8) translation: 2 units down
D
(
−4, 1
)
, A
(
−2, 5
)
, S
(
−1, 4
)
, N
(
−1, 2
)
9) translation: 4 units left and 4 units up
J
(
−1, −2
)
, A
(
−1, 0
)
, N
(
3, −3
)
10) translation: 3 units right and 4 units up
Z
(
−4, −3
)
, I
(
−2, −2
)
, V
(
−2, −4
)
Write a rule to describe each transformation.
11)
x
y
I
Q
U
M
I'
Q'
U'
M'
12)
x
y
K
F
D
I
K'
F'
D'
I'
13)
x
y
E
L
B
E'
L'
B'
14)
x
y
P
F
C
P'
F'
C'
-2-
5
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8
Worksheet by Kuta Software LLC
Kuta Software - Infinite Geometry Name___________________________________
Period____Date________________
Translations
Graph the image of the figure using the transformation given.
1) translation: 5 units right and 1 unit up
x
y
B
G
T
2) translation: 1 unit left and 2 units up
x
y
M
Y
G
3) translation: 3 units down
x
y
U
Q
L
4) translation: 5 units right and 2 units up
x
y
I
X
E
5) translation: 4 units right and 4 units down
x
y
A
J
I
6) translation: 2 units right and 3 units up
x
y
E
J
M
X
-1-
6
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H
.
z
Worksheet by Kuta Software LLC
Write a rule to describe each transformation.
7)
x
y
K
Z
I
K'
Z'
I'
8)
x
y
I
J
V
T
I'
J'
V'
T'
9)
x
y
N
U
H
N'
U'
H'
10)
x
y
L
A
P
L'
A'
P'
11)
x
y
N
H
Y
W
N'
H'
Y'
W'
12)
x
y
P
T
B
P'
T'
B'
-2-
7
4
8
5
Lesson 2 Problem-Solving Practice
Reflections
1. DESIGNS Half of a design is shown below. Reflect the
figure across the x-axis to obtain the completed design.
2. DESIGNS Half of a design is shown below. Reflect the
figure across the y-axis to obtain the completed design.
3. LOGO Half of a logo is shown below. Reflect the figure
across the y-axis to obtain the completed figure.
4. SYMBOLS The figure shows a ray plotted on a coordinate
system. Reflect the ray across the x-axis. Graph the reflected
image.
5. ARCHITECTURE A corporate plaza is to be built around a
small lake. Building 1 has already been built. Suppose there
are axes through the lake as shown. Show where Building 2
should be built if it will be a reflection of Building 1 across
the y-axis followed by a reflection across the x-axis.
6. ARCHITECTURE Use the information from Exercise 5.
Suppose that a third building is to be built as shown. To
complete the business park, show where a fourth building
should be built if it is a reflection of Building 3 across the x
and y-axis.
9
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4
Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Reflections of Shapes
Graph the image of the figure using the transformation given.
1) reflection across the x-axis
x
y
L
G
Q
2) reflection across
y = 3
x
y
L
U
X
3) reflection across
y = 1
x
y
I
T
Y
4) reflection across the x-axis
x
y
M
Z
D
P
5) reflection across the x-axis
T
(
2, 2
)
, C
(
2, 5
)
, Z
(
5, 4
)
, F
(
5, 0
)
x
y
6) reflection across
y = −2
H
(
−1, −5
)
, M
(
−1, −4
)
, B
(
1, −2
)
, C
(
3, −3
)
x
y
-1-
10
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Worksheet by Kuta Software LLC
Find the coordinates of the vertices of each figure after the given transformation.
7) reflection across the x-axis
K
(
1, −1
)
, N
(
4, 0
)
, Q
(
4, −4
)
8) reflection across
y = −1
R
(
−3, −5
)
, N
(
−4, 0
)
, V
(
−2, −1
)
, E
(
0, −4
)
9) reflection across
x = 3
F
(
2, 2
)
, W
(
2, 5
)
, K
(
3, 2
)
10) reflection across
x = −1
V
(
−3, −1
)
, Z
(
−3, 2
)
, G
(
−1, 3
)
, M
(
1, 1
)
Write a rule to describe each transformation.
11)
x
y
K
I
H
I'
H'
K'
12)
x
y
G
X
F
X'
F'
G'
13)
x
y
N
Z
X
Z'
X'
N'
14)
x
y
U
B
M
S
B'
M'
S'
U'
-2-
11
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Geometry Name___________________________________
Period____Date________________
Reflections
Graph the image of the figure using the transformation given.
1) reflection across
y = −2
x
y
E
I
Q
Z
2) reflection across the x-axis
x
y
W
M
D
A
3) reflection across
y = −
x
x
y
J
A
S
T
4) reflection across
y = −1
x
y
B
I
W
L
5) reflection across
x = −3
x
y
P
I
W
S
6) reflection across
y =
x
x
y
Q
H
L
P
-1-
12
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Worksheet by Kuta Software LLC
Write a rule to describe each transformation.
7)
x
y
J
V
F
K
V'
F'
K'
J'
8)
x
y
U
T
V
D
T'
V'
D'
U'
9)
x
y
A
W
R
G
W'
R'
G'
A'
10)
x
y
R
K
P
K'
P'
R'
11)
x
y
A
P
U
H
P'
U'
H'
A'
12)
x
y
Z
D
H
U
D'
H'
U'
Z'
-2-
13
6
Lesson 3 Skills Practice
Rotations
For Exercises 1 and 2, graph ∆XYZ and its image after each rotation. Then give the coordinates of the vertices for
∆XʹYʹZʹ.
1. 180° clockwise about vertex Z
2. 90° clockwise about vertex X
3. Triangle JKL has vertices J(-4, 4), K(-1, 3), and L(-2, 1). Graph the figure
and its rotated image after a clockwise rotation of 90° about the origin. Then
give the coordinates of the vertices for triangle J'K'L'.
4. Quadrilateral BCDE has vertices B(3, 6), C(6, 5), D(5, 2), and E(2, 3). Graph
the figure and its rotated image after a counterclockwise rotation of 180°
about the origin. Then give the coordinates of the vertices for quadrilateral
BʹCʹDʹE'.
14
7
Lesson 3 Problem-Solving Practice
Rotations
1. OPEN-ENDED Draw a figure that has rotational symmetry
with 90° and 180° as its angles of rotation.
2. CLASSIFY Identify the transformation shown below as a
translation, reflection, or rotation. Explain.
3. ROTATIONS Which figure below was rotated 90°
counterclockwise?
4. LETTERS Which capital letters in the word
TRANSFORMATION produce the same letter after being
rotated 180°?
5. REAL-WORLD Describe a real-world example of where you
could find a rotation.
6. ART An art design is shown. State the angles of rotation.
15
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W
Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Rotations of Shapes
Graph the image of the figure using the transformation given.
1) rotation 180° about the origin
x
y
J
Q
H
2) rotation 90° counterclockwise about the
origin
x
y
S
B
L
3) rotation 90° clockwise about the origin
x
y
M
B
F
H
4) rotation 180° about the origin
x
y
U
H
F
5) rotation 90° clockwise about the origin
U
(
1, −2
)
, W
(
0, 2
)
, K
(
3, 2
)
, G
(
3, −3
)
x
y
6) rotation 180° about the origin
V
(
2, 0
)
, S
(
1, 3
)
, G
(
5, 0
)
x
y
-1-
16
©
b
3
2
G
0
p
1
f
1
s
N
K
D
u
t
t
o
a
t
Q
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Find the coordinates of the vertices of each figure after the given transformation.
7) rotation 180° about the origin
Z
(
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(
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(
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9) rotation 90° clockwise about the origin
S
(
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10) rotation 180° about the origin
V
(
−5, −3
)
, A
(
−3, 1
)
, G
(
0, −3
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Write a rule to describe each transformation.
11)
x
y
Q
N
R
E
Q'
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12)
x
y
S
U
X
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Geometry Name___________________________________
Period____Date________________
Rotations
Graph the image of the figure using the transformation given.
1) rotation 180° about the origin
x
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2) rotation 180° about the origin
x
y
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3) rotation 90° counterclockwise about the
origin
x
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4) rotation 90° clockwise about the origin
x
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5) rotation 90° clockwise about the origin
x
y
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6) rotation 180° about the origin
x
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18
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Write a rule to describe each transformation.
7)
x
y
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H'
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8)
x
y
Z
N
K
A
Z'
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9)
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10)
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12)
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R'
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19
8
Lesson 4 Skills Practice
Dilations
Find the coordinates of the vertices of each figure after a dilation with the given scale factor k. Then graph the original
image and the dilation.
1. J(–4, –1), K(0, 4), L(–4, –2);
2. R(–2, 1), A(1, 1), I(0, –1), N(–1, –1); k = 2
3. P(–3, 3), Q(6, 3), R(6, –3), S(–3, –3);
4. A(2, 1), B(3, 0), C(1, –2); k = 3
5. PHOTOS Kiesha used a photo that measured 4 inches by 6 inches to make a copy that measured 8 inches by 12 inches.
What is the scale factor of the dilation?
6. MODELS David built a model of a regulation basketball court. His model measured approximately 3.75 feet long by 2 feet
wide. The dimensions of a regulation court are 94 feet long by 50 feet wide. What is the scale factor David used to build his
model?
7. BLUEPRINTS On the blueprints of Mr. Wong’s house, his great room measures 4.5 inches by 5 inches. The actual great
room measures 18 feet by 20 feet. What is the scale factor of the dilation?
20
9
Lesson 4 Problem-Solving Practice
Dilations
1. GEOMETRY Find the coordinates of the triangle shown
below after a dilation with a scale factor of 4.
2. PHOTOS Daniel is using a scale factor of 10 to enlarge a
class photo that measures 3.5 inches by 5 inches. What are
the dimensions of the photo after the dilation?
3. DOGS Isabel has a mother dog and her puppy that look
exactly alike. The puppy weighs 6 pounds, and the mother
weighs 48 pounds. Assuming the two dogs are similar, what
is the scale factor of the dilation?
4. GEOMETRY Find the coordinates of the quadrilateral
shown below after a dilation with a scale factor of
.
5. BLUEPRINTS Abby’s family is building a new house. On
the blueprints of the house, Abby’s bedroom measures 3
inches by 3.75 inches. Her actual bedroom will measure 8
feet by 10 feet. What is the scale factor for the dilation?
6. ART William saw a painting in a museum, and later found a
picture of that same painting in a book. The actual painting
measured 36 inches by 54 inches. The picture of the painting
measured 4 inches by 6 inches. What is the scale factor for
the dilation?
21
10
Lesson 1 Skills Practice
Congruence and Transformations
Determine if the two figures are congruent by using transformations.
Explain your reasoning.
1. 2.
3. 4.
5. 6.
22
11
Lesson 1 Problem-Solving Practice
Congruence and Transformations
Determine if the two figures are congruent by using transformations. Explain your reasoning.
1.
2.
3. The community softball team has created the following logo
for their jerseys. What transformations could be used if the
letter “M” is the image and the letter “W” is the preimage?
Are the two figures congruent? Explain.
4. For the local art gallery opening, the curator had the design
shown below created. What transformations could be used if
the white figure is the image and the black figure is the
preimage? Are the two figures congruent? Explain.
5. For his school web page, Manuel created the logo shown at the right. What
transformations could be used if the gray figure is the preimage and the
black figure is the image? Are the two figures congruent? Explain.
23
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Geometry Name___________________________________
Period____Date________________
All Transformations
Graph the image of the figure using the transformation given.
1) rotation 90° counterclockwise about the
origin
x
y
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Z
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2) translation: 4 units right and 1 unit down
x
y
Y
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3) translation: 1 unit right and 1 unit up
x
y
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4) reflection across the x-axis
x
y
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Write a rule to describe each transformation.
5)
x
y
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6)
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7)
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x
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Graph the image of the figure using the transformation given.
9) rotation 90° clockwise about the origin
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(
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x
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10) reflection across
y =
x
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(
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, J
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x
y
Find the coordinates of the vertices of each figure after the given transformation.
11) rotation 180° about the origin
E
(
2, −2
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, J
(
1, 2
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(
3, 3
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y = 2
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1, 5
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, C
(
3, 2
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13) translation: 7 units right and 1 unit down
J
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14) translation: 6 units right and 3 units down
S
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-2-
25
12
Lesson 2 Skills Practice
Congruence
Write congruence statements comparing the corresponding parts in each set of congruent figures.
1. 2.
3. 4.
5. 6.
26
13
Lesson 2 Problem-Solving Practice
Congruence
1. In the quilt design shown, ∆RST ∆RWX. What is the
measure of STR?
2. In the roof construction shown, ∆CBD ∆EFD. If CB = 11
feet, what is EF?
3. In the stage truss shown below, ∆HJK ∆HLK. If LHK =
71°, what is the measure of JHK?
4. Triangle FGH is congruent to ∆PQR. Write congruence
statements comparing the corresponding parts. Then
determine which transformations map ∆FGH onto ∆PQR.
5. In the baseball diamond shown, ∆BEA ∆ARB. If BE = 90
feet, what is AR?
6. Parallelograms ABCD and FGHI are congruent. If AB = 64
centimeters, what is FG?
27
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Lesson 3 Skills Practice
Similarity and Transformations
Determine if the two figures are similar by using transformations.
Explain your reasoning.
1. 2.
3. 4.
28
15
Lesson 3 Problem-Solving Practice
Similarity and Transformations
1. Stephanie has a photo of her family that she is placing in a
frame. The original photo is 5 inches by 7 inches. She
enlarges the photo by a scale factor of 2 to place in her room.
She then enlarges this photo by a scale factor of 1.5 to place
above her fireplace. What are the dimensions of the photo
above her fireplace? Are the enlarged photos similar to the
original?
2. An architect is designing a decorative window. The window
uses similar parallelograms. If parallelogram ABEG is
similar to parallelogram ACDF, what is the length of AF?
3. An iron-on measures 3 inches by 4 inches. It is enlarged by a
scale factor of 2 for a t-shirt. The second iron-on is enlarged
by a scale factor of 3 for a bag. What are the dimensions of
the largest iron on? Are both of the enlarged iron-ons similar
to the original?
4. Casey is reducing the size of her painting to make it into a
postcard. The painting is 12 inches by 20 inches. She will
reduce it by a scale factor of
What are the dimensions of
the postcard?
5. Ryan is using tiles in his bathroom. He chooses 1-inch by 2-
inch tiles for the border and would like tiles that are similar
to the border as the interior tiles. The interior tiles will be
larger by a scale factor of 3.5. What are the dimensions of
the interior tiles?
6. For an art show, an artist is projecting a piece of art 5 inches
by 7 inches onto a white wall. It will be enlarged by a scale
factor of 12. What are the dimensions of the art on the wall?
29
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Lesson 5 Skills Practice
Similar Triangles and Indirect Measurement
In Exercises 1– 6, the triangles are similar. Write a proportion and solve the problem.
1. HEIGHT How tall is Becky? 2. FLAGS How tall is the flagpole?
3. BEACH How deep is the water 50 feet 4. ACCESSIBILITY How high is the ramp
from shore? when it is 2 feet from the building?
(Hint: ∆ABE ~ ∆ACD)
5. AMUSEMENT PARKS How far is the water 6. CLASS CHANGES How far is the entrance
ride from the roller coaster? Round to to the gymnasium from the band room?
the nearest tenth.
30
17
Lesson 5 Problem-Solving Practice
Similar Triangles and Indirect Measurement
1. HEIGHT Eduardo is 6 feet tall and casts a 12-foot shadow.
At the same time, Diane casts an 11-foot shadow. How tall is
Diane?
2. LIGHTING If a 25-foot-tall house casts a 75-foot shadow at
the same time that a streetlight casts a 60-foot shadow, how
tall is the streetlight?
3. FLAGPOLE Lena is
feet tall and casts an 8-foot shadow.
At the same time, a flagpole casts a 48-foot shadow. How tall
is the flagpole?
4. LANDMARKS A woman who is 5 feet 5 inches tall is
standing near the Space Needle in Seattle, Washington. She
casts a 13-inch shadow at the same time that the Space
Needle casts a 121-foot shadow. How tall is the Space
Needle?
5. NATIONAL MONUMENTS A 42-foot flagpole near the
Washington Monument casts a shadow that is 14 feet long.
At the same time, the Washington Monument casts a shadow
that is 185 feet long. How tall is the Washington Monument?
6. ACCESSIBILITY A ramp slopes upward from the sidewalk
to the entrance of a building at a constant incline. If the
ramp is 2 feet high when it is 5 feet from the sidewalk, how
high is the ramp when it is 7 feet from the sidewalk?
31
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Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Similar Figures
Each pair of figures is similar. Find the missing side.
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2)
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32
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33
18
Lesson 6 Skills Practice
Slope and Similar Triangles
Graph each pair of similar triangles. Then write a proportion comparing the rise to the run for each of the similar slope
triangles and find the numeric value.
1. ∆CDE with vertices C(–6, –3), 2. ∆RST with vertices R(–4, 5), S(–4, –4),
D(–3, –2), and E(–3, –3); ∆MNO with and T(2, –4); ∆UVW with vertices
vertices M(0, –1), N(6, 1), and O(6, –1) U(–2, 2), V(–2, –1), and W(0, –1)
3. ∆QRP with vertices Q(–5, 1), R(–1, 3), 4. ∆CAM with vertices at C(–1, 6), A(–1, 3),
and P(–1, 1); ∆RKJ with vertices and M(0, 3); ∆CEN with vertices at
R(–1, 3), K(5, 6), and J(5, 3). C(–1, 6), E(–1, –3), and N(2, –3)
34
19
Lesson 6 Problem-Solving Practice
Slope and Similar Triangles
1. The slope of a roof line is also called the pitch. Find the pitch
of the roof shown.
2. A carpenter is building a set of steps for a bunk bed. The
plan for the steps is shown below. Using points A and B,
find the slope of the line up the steps. Then verify that the
slope is the same at a different location by choosing a
different set of points.
3. A ladder is leaning up against the side of a house. Use two
points to find the slope of the ladder. Then verify that the
slope is the same at a different location by choosing a
different set of points.
4. The graph shows the plans for a bean bag tossing game. Use
two points to find the slope of the game. Then verify that
the slope is the same at a different location by choosing a
different set of points.
35
20
Lesson 5 Skills Practice
The Pythagorean Theorem
Write an equation you could use to find the length of the missing side of each right triangle. Then
find the missing length. Round to the nearest tenth if necessary.
1. 2. 3.
4. 5. 6.
7. a = 1 m, b = 3 m 8. a = 2 in., c = 5 in.
9. b = 4 ft, c = 7 ft 10. a = 4 km, b = 9 km
11. a = 10 yd, c = 18 yd 12. b = 18 ft, c = 20 ft
13. a = 5 yd, b = 11 yd 14. a = 12 cm, c = 16 cm
15. b = 22 m, c = 25 m 16. a = 21 ft, b = 72 ft
17. a = 36 yd, c = 60 yd 18. b = 25 mm, c = 65 mm
Determine whether each triangle with sides of given lengths is a right triangle. Justify your
answer.
19. 10 yd, 15 yd, 20 yd 20. 21 ft, 28 ft, 35 ft
21. 7 cm, 14 cm, 16 cm 22. 40 m, 42 m, 58 m
23. 24 in., 32 in., 38 in. 24. 15 mm, 18 mm, 24 mm
36
21
Lesson 5 Problem-Solving Practice
The Pythagorean Theorem
1. ART What is the length of a diagonal of a
rectangular picture whose sides are 12 inches
by 17 inches? Round to the nearest tenth of
an inch.
2. GARDENING Ross has a rectangular
garden in his back yard. He measures one
side of the garden as 22 feet and the
diagonal as 33 feet. What is the length of
the other side of his garden? Round to the
nearest tenth of a foot.
3. TRAVEL Troy drove 8 miles due east and
then 5 miles due north. How far is Troy from
his starting point? Round the answer to the
nearest tenth of a mile.
4. GEOMETRY What is the perimeter of a
right triangle if the hypotenuse is 15
centimeters and one of the legs is 9
centimeters?
5. ART Anna is building a rectangular picture
frame. If the sides of the frame are 20 inches
by 30 inches, what should be the diagonal
measure? Round to the nearest tenth of an
inch.
6. CONSTRUCTION A 20-foot ladder leaning
against a wall is used to reach a window
that is 17 feet above the ground. How far
from the wall is the bottom of the ladder?
Round to the nearest tenth of a foot.
7. CONSTRUCTION A door frame is 80
inches tall and 36 inches wide. What is the
length of a diagonal of the door frame?
Round to the nearest tenth of an inch.
8. TRAVEL Tina measures the distances
between three cities on a map. The
distances between the three cities are 45
miles, 56 miles, and 72 miles. Do the
positions of the three cities form a right
triangle?
37
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
The Pythagorean Theorem
Do the following lengths form a right triangle?
1)
6
8
9
2)
5
12
13
3)
6
8
10
4)
3
4
5
5) a = 6.4, b = 12, c = 12.2 6) a = 2.1, b = 7.2, c = 7.5
Find each missing length to the nearest tenth.
7)
4
8
8)
6
3
9)
7
10
10)
7
3
11)
7
2
12)
2
6
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Kuta Software - Infinite Geometry Name___________________________________
Period____Date________________
The Pythagorean Theorem and Its Converse
Find the missing side of each triangle. Round your answers to the nearest tenth if necessary.
1)
x
12 in
13 in
2)
3 mi
4 mi
x
3)
11.9 km
x
14.7 km
4)
6.3 mi
x
15.4 mi
Find the missing side of each triangle. Leave your answers in simplest radical form.
5)
x
13 yd
15 yd
6)
8 km
x
16 km
Find the missing side of each right triangle. Side
c is the hypotenuse. Sides
a and
b are the legs. Leave
your answers in simplest radical form.
7)
a = 11 m,
c = 15 m
8)
b = 6 yd,
c = 4 yd
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State if each triangle is a right triangle.
9)
9 m
12 m
15 m
10)
2 39 ft
10 ft
16 ft
11)
115 yd
9 yd
11 yd
12)
32.5 ft
39 ft
48.5 ft
State if the three sides lengths form a right triangle.
13) 10 cm, 49.5 cm, 50.5 cm 14) 9 in, 12 in, 15 in
State if each triangle is acute, obtuse, or right.
15)
9 cm
12 cm
17 cm
16)
9.6 in
18 in
20.1 in
State if the three side lengths form an acute, obtuse, or right triangle.
17) 6 mi,
2 55 mi, 17 mi
18) 4.8 km, 28.6 km, 29 km
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41
22
Lesson 6 Skills Practice
Use the Pythagorean Theorem
Write an equation that can be used to answer the question. Then solve.
Round to the nearest tenth if necessary.
1. How far apart are the spider and 2. How long is the tabletop?
the fly?
3. How high will the ladder reach? 4. How high is the ramp?
5. How far apart are the two cities? 6. How far is the bear from camp?
7. How tall is the table? 8. How far is it across the pond
42
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Lesson 6 Problem-Solving Practice
Use the Pythagorean Theorem
1. RECREATION A pool table is 8 feet long
and 4 feet wide. How far is it from one
corner pocket to the diagonally opposite
corner pocket? Round to the nearest tenth.
2. TRIATHLON The course for a local
triathlon has the shape of a right triangle.
The legs of the triangle consist of a 4-mile
swim and a 1l mile run. The hypotenuse of
the triangle is the biking portion of the
event. How far is the biking part of the
triathlon? Round to the nearest tenth if
necessary.
3. LADDER A ladder 17 feet long is leaning
against a wall. The bottom of the ladder is
8 feet from the base of the wall. How far
up the wall is the top of the ladder? Round
to the nearest tenth if necessary.
4. TRAVEL Tara drives due north for 22
miles then east for 11 miles. How far is
Tara from her starting point? Round to the
nearest tenth if necessary.
5. FLAGPOLE A wire 3l feet long is
stretched from the top of a flagpole to the
ground at a point 15 feet from the base of
the pole. How high is the flagpole? Round
to the nearest tenth if necessary.
6. ENTERTAINMENT Isaac's television is
25 inches wide and 18 inches high. What is
the diagonal size of Isaac's television?
Round to the nearest tenth if necessary.
43
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Geometry Name___________________________________
Period____Date________________
Multi-Step Pythagorean Theorem Problems
Find the area of each triangle. Round intermediate values to the nearest tenth. Use the rounded values
to calculate the next value. Round your final answer to the nearest tenth.
1)
9
8
2)
9
7
3)
6
8
4)
7
8
5)
5 5
7.6
6)
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5.2
7)
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8
8)
7
5
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9)
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11)
29
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13)
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16)
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9
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45
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Lesson 7 Skills Practice
Distance on the Coordinate Plane
Find the distance between each pair of points whose coordinates are given.
Round to the nearest tenth if necessary.
1. 2. 3.
4. 5. 6.
Graph each pair of ordered pairs. Then find the distance between the points.
Round to the nearest tenth if necessary.
7. (–3, 0), (3, –2) 8. (–4, –3), (2, 1) 9. (0, 2), (5, –2)
10. (–2, 1), (–1, 2) 11. (0, 0), (–4, –3)
46
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Lesson 7 Problem-Solving Practice
Distance on the Coordinate Plane
1. ARCHAEOLOGY An archaeologist at a
dig sets up a coordinate system using string.
Two similar artifacts are found—one at
position (1, 4) and the other at (5, 2). How
far apart were the two artifacts? Round to
the nearest tenth of a unit if necessary.
2. GARDENING Vega set up a coordinate
system with units of feet to locate the
position of the vegetables she planted in
her garden. She has a tomato plant at (1, 3)
and a pepper plant at (5, 6). How far apart
are the two plants? Round to the nearest
tenth if necessary.
3. CHESS April is an avid chess player. She
sets up a coordinate system on her chess
board so she can record the position of
the pieces during a game. In a recent
game, April noted that her king was at
(4, 2) at the same time that her opponent's
king was at (7, 8). How far apart were the
two kings? Round to the nearest tenth of
a unit if necessary.
4. MAPPING Cory makes a map of his
favorite park, using a coordinate system
with units of yards. The old oak tree is at
position (4, 8) and the granite boulder is at
position (–3, 7). How far apart are the old
oak tree and the granite boulder? Round to
the nearest tenth if necessary
5. TREASURE HUNTING Taro uses a
coordinate system with units of feet to keep
track of the locations of any objects he
finds with his metal detector. One lucky
day he found a ring at (5, 7) and an old coin
at (10, 19). How far apart were the ring and
coin before Taro found them? Round to the
nearest tenth if necessary.
6. GEOMETRY The coordinates of points A
and B are (–7, 5) and (4, –3), respectively.
What is the distance between the points,
rounded to the nearest tenth?
7. GEOMETRY The coordinates of points A,
B, and C are (5, 4), (–2, 1), and (4, –4),
respectively. Which point, B or C, is closer
to point A?
8. THEME PARK Bryce is looking at a map
of a theme park. The map is laid out in a
coordinate system. Bryce is at (2, 3). The
roller coaster is at (7, 8), and the water ride
is at (9, 1). Is Bryce closer to the roller
coaster or the water ride?
47
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Lesson 1 Skills Practice
Volume of Cylinders
Find the volume of each cylinder. Round to the nearest tenth.
1. 2. 3.
4. 5. 6.
7. radius = 8.8 cm 8. radius = 4 ft
height = 4.7 cm height =
ft
9. diameter = 10 mm 10. diameter = 7.1 in.
height = 4 mm height = 1 in.
11. diameter = 12 ft 12. diameter =
in.
height = 18 ft height = 5 in.
48
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Lesson 1 Problem-Solving Practice
Volume of Cylinders
1. WATER STORAGE A cylindrical water tank has a
diameter of 5.3 meters and a height of 9 meters. What is the
maximum volume that the water tank can hold? Round to the
nearest tenth.
2. PACKAGING A can of corn has a diameter of 6.6
centimeters and a height of 9.9 centimeters. How much corn
can the can hold? Round to the nearest tenth.
3. CONTAINERS Felisa wants to determine the maximum
capacity of a cylindrical bucket that has a radius of 6 inches
and a height of 12 inches. What is the capacity of Felisa’s
bucket? Round to the nearest tenth.
4. GLASS Antoine is designing a new, cylindrical drinking
glass. If the glass has a diameter of 8 centimeters and a
height of 12.8 centimeters, what is its volume? Round to the
nearest tenth.
5. PAINT A can of paint is 15 centimeters high and has a
diameter of 13.6 cm. What is the volume of the can? Round
to the nearest tenth.
6. SPICES A spice manufacturer uses a cylindrical dispenser
like the one shown. Find the volume of the dispenser to the
nearest tenth.
49
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Lesson 2 Skills Practice
Volume of Cones
Find the volume of each cone. Round to the nearest tenth.
1.
2.
3.
4.
5. diameter: 10 centimeters; height: 14 centimeters
6. radius: 8.7 feet; height: 16 feet
7. height: 34 centimeters; diameter: 6 centimeters
8. FUNNEL A funnel is in the shape of a cone. The radius is 2 inches and the height is 4.6 inches. Find the volume of the
funnel. Round to the nearest tenth.
50
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Lesson 2 Problem-Solving Practice
Volume of Cones
1. DESSERT Find the volume of the ice cream cone shown
below. Round to the nearest tenth.
2. SALT Lecretia uses a small funnel as shown below to fill
her salt shaker. Find the volume of the funnel. Round to the
nearest tenth.
3. ENTRYWAY The top of the stone posts at the entry to an
estate are in the shape of a cone as shown below. Find the
volume of stone needed to make the top of the post. Round
to the nearest tenth.
4. PAPERWEIGHT Marta bought a paperweight in the shape
of a cone. The radius was 10 centimeters and the height 9
centimeters. Find the volume. Round to the nearest tenth.
5. LAMPSHADE A lampshade is in the shape of a cone. The
diameter is 5 inches and the height 6.5 inches. Find the
volume. Round to the nearest tenth.
6. CANDY A piece of candy is in the shape of a cone. The
height of the candy is 2 centimeters and the diameter is 1
centimeter. Find the volume. Round to the nearest tenth.
51
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Lesson 3 Skills Practice
Volume of Spheres
Find the volume of each sphere. Round to the nearest tenth.
1. 2.
3. 4.
Find the volume of each hemisphere. Round to the nearest tenth.
5. 6.
7. 8.
52
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Lesson 3 Problem-Solving Practice
Volume of Spheres
1. DESSERT A scoop of ice cream is in the shape of a sphere.
The diameter of the scoop of ice cream is 2.5 inches. Find the
volume of the ice cream. Round to the nearest tenth.
2. TOYS A playground ball has a radius of 7.5 inches. Find the
volume of the ball. Round to the nearest tenth.
3. GLOBE A globe has a diameter of 14 inches. Find the
volume of the globe. Round to the nearest tenth.
4. JEWELRY Jackie is using spherical beads to create a border
on a picture frame. Each bead has a diameter of 1.5
millimeters. Find the volume of each bead. Round to the
nearest tenth.
5. DECORATION A glass ball is used to decorate a garden.
The radius of the ball is 25 centimeters. Find the volume.
Round to the nearest tenth.
6. BALLOONS Mrs. McCullough is purchasing balloons for a
party. Each spherical balloon is inflated with helium. How
much helium is in the balloon if the balloon has a radius of 9
centimeters? Round to the nearest tenth.
53
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Volumes of Solids
Find the volume of each figure. Round to the nearest tenth.
1)
2 yd
1.5 yd
4 yd
5 yd
4 yd
2)
5 mi
3 mi
4 mi
5 mi
3)
3 yd
3 yd
5 yd
4)
3 km
2 km
5)
3 in
4 in
6)
2 m
2 m
2 m
2 m
2 m
7)
2.5 yd
6 yd
5 yd
3 yd
3 yd
8)
1 in
2 in
1 in
-1-
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9)
12 cm
9 cm
15 cm
19 cm
10)
17 m
4 m
4 m
17 m
10 m
11)
12 m
15 m
20 m
12)
9.3 m
10 m
18 m
19 m
15 m
13)
11 mi
40 mi
14)
11 yd
14 yd
15)
19 in
11 in
11 in
19 in
4 in
16)
17 mi
18 mi
17 mi
17) A cylinder with a radius of 3 cm and a height
of 7 cm.
18) A cone with diameter 20 cm and a height of 20
cm.
19) A cone with diameter 14 yd and a height of 14
yd.
20) A rectangular prism measuring 10 m and 7 m
along the base and 12 m tall.
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55
