Download UNIT 5

UNIT 5
Relationships
in Geometry
MODULE
MODULE
15
15
Angles, Triangles,
and Equations
6.8.A
16
16
Area and Volume
MODULE
MODULE
Equations
6.8.B, 6.8.C, 6.8.D
CAREERS IN MATH
Theater Set Construction A person
who works in theater set construction works
with the set designer to create scenery and
needs technical precision when scaling and
building sets based on the dimensions of the
models.
If you are interested in a career in theater
set construction, you should study these
mathematical subjects:
• Geometry
• Algebra
• Trigonometry
At the end of the unit,
check out how theater set
construction workers
use math.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Nik Wheeler/Corbis
Research other careers that require technical
precision in scaling and building models.
Unit 5 Performance Task
Unit 5
417
Preview
UNIT 5
Vocabulary
Use the puzzle to preview key vocabulary from this unit. Unscramble
the circled letters within found words to answer the riddle at the
bottom of the page.
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• A triangle that has three congruent sides and three congruent angles. (Lesson 15-2)
• A triangle that has two congruent sides. (Lesson 15-3)
• A triangle that contains a right angle. (Lesson 15-2)
• A quadrilateral where opposite sides are congruent and parallel. (Lesson 16-1)
• A quadrilateral in which all sides are congruent and opposite sides are parallel. (Lesson 16-1)
Q:
Where does a mathematician go when
she commits a crime?
A:
418
Vocabulary Preview
!
© Houghton Mifflin Harcourt Publishing Company
• A triangle that has no congruent sides. (Lesson 15-3)
Angles, Triangles,
and Equations
?
MODULE
15
LESSON 15.1
ESSENTIAL QUESTION
Determining When
Three Lengths Form
a Triangle
How can you use angles,
triangles, and equations
to solve real-world
problems?
6.8.A
LESSON 15.2
Sum of Angle
Measures in a Triangle
6.8.A
LESSON 15.3
Relationships
Between Sides and
Angles in a Triangle
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Richard Nowitz/
Photodisc/Getty Images
6.8.A
Real-World Video
You can find examples of triangles all around you.
Some buildings, such as the Transamerica Tower,
have triangular faces.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
419
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this chapter.
Inverse Operations
EXAMPLE
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7k = 35
Online
Assessment and
Intervention
k is multiplied by 7.
To solve the equation, use the
inverse operation, division.
35
7k
__
= __
7
7
k=5
7 is added to k.
k+7=9
k+7-7=9-7
k= 2
To solve the equation, use the inverse
operation, subtraction.
Solve each equation using the inverse operation.
1. 9p = 54
2. m - 15 = 9
3. __b8 = 4
4. z + 17 = 23
Name Angles
EXAMPLE
Use three points of an angle, including the vertex,
to name the angle. If there is only one angle at the
vertex, you can name the angle by the vertex.
Write the vertex between the other two points.
∠AMG, ∠GMA, or ∠M.
A
M
G
5.
6.
Z
7.
N
B
S
K
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Unit 5
T
J
V
M
R
P
B
T
F
L
© Houghton Mifflin Harcourt Publishing Company
Give two names for the angle formed by the dashed rays.
Reading Start-Up
Vocabulary
Review Words
✔ acute angle (ángulo
agudo)
angle (ángulo)
equilateral triangle
(triángulo equilátero)
inequalities (desigualdad)
line segments (segmentos
de línea)
✔ obtuse angle (ángulo
obtuso)
✔ right angle (ángulo recto)
right triangle (triángulo
rectángulo)
vertex (vértice)
Visualize Vocabulary
Use the ✔ words to complete the graphic. You will put one
word in each oval.
Types of Angles
Description
Angle
angle measure
> 0° and < 90°
angle measure
> 90° and < 180°
angle measure = 90°
Understand Vocabulary
Complete the sentences using the review words.
1. A triangle that contains a right angle is a
2. An
congruent angles.
has three congruent sides and three
3. The sides of triangles are
© Houghton Mifflin Harcourt Publishing Company
.
meet to form an angle of a triangle is called a
. Where two lines
.
Active Reading
Pyramid Before beginning the module,
create a pyramid to help you organize what
you learn. Label each side with one of the
lesson titles from this module. As you study
each lesson, write important ideas like
vocabulary, properties, and formulas on the
appropriate side.
Module 15
421
MODULE 15
Unpacking the TEKS
Understanding the TEKS and the vocabulary terms in the TEKS
will help you know exactly what you are expected to learn in this
module.
6.8.A
Extend previous knowledge of
triangles and their properties
to include the sum of angles
of a triangle, the relationship
between the lengths of sides
and measures of angles in a
triangle, and determining
when three lengths form a
triangle.
What It Means to You
You will learn to determine if three
lengths can form a triangle.
UNPACKING EXAMPLE 6.8.A
A map of a new dog park shows that it is
triangular and that the sides measure 18
yd, 37 yd, and 17 yd. Are the dimensions
possible? Explain your reasoning.
Find the sum of the lengths of each pair of sides.
Compare the sum to the third side.
?
?
?
18 + 37 > 17
18 + 17 > 37
37 + 17 > 18
55 > 17 ✔
35 ≯ 34 ✘
54 > 18 ✔
6.8.A
Extend previous knowledge of
triangles and their properties
to include the sum of angles
of a triangle, the relationship
between the lengths of sides
and measures of angles in a
triangle, and determining when
three lengths form a triangle.
What It Means to You
You will learn how to find the measure of an angle of a triangle if
you know the measures of the other two angles.
The measures of two of the angles of a triangle are 47° and 81°.
What is the measure of the third angle of the triangle?
x
47° + 81° +
128°
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to see all
the
unpacked.
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422
Unit 5
C
m∠A + m∠B + m∠C = 180°
+
x
= 180°
x = 180°
x = 52°
47°
A
The third angle of the triangle measures 52°.
81°
B
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Thinkstock/
Comstock Images/Getty Images
The sum of two of the given lengths is not greater than the third
length. So, the dog park cannot have these side lengths.
LESSON
15.1
?
Determining When
Three Lengths Form
a Triangle
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—
6.8.A Extend previous
knowledge of triangles
and their properties to
include . . . determining when
three lengths form a triangle.
How can you use the relationship between side lengths to
determine when three lengths form a triangle?
EXPLORE ACTIVITY
6.8.A
Drawing Three Sides
Use geometry software to draw a triangle whose sides
have the following lengths: 2 units, 3 units, and 4 units.
E
F
c=4
A Draw three line segments of 2, 3, and 4 units of length.
C
D
b=3
A
B
a=2
___
B Let AB be the base of the triangle. Place endpoint
C on top of endpoint B and endpoint E on top of
endpoint A. These will become two of the vertices
of the triangle.
F
D
c=4
E
B
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A a=2 C
C Using the endpoints C and E as fixed vertices, rotate
endpoints F and D to see if they will meet in a single point.
The line segments of 2, 3, and 4 units do / do not
form a triangle.
D Repeat Steps 2 and 3, but start with a different base
length. Do the line segments make the exact same
triangle as the original?
D
c=4
E
A a=2 C
b=3
F
b=3
B
The line segments do / do not make the same triangle as the
original.
E Draw three line segments of 2, 3, and 6 units. Can you form
a triangle with the given segments?
The line segments of 2, 3, and 6 units do / do not form a triangle.
Lesson 15.1
423
EXPLORE ACTIVITY (cont’d)
Reflect
1.
Conjecture Try to make triangles using real world objects such as
three straws of different lengths. Find three side lengths that form a
triangle and three side lengths that do not form a triangle. What do
you notice about the lengths that do not form a triangle?
Using Triangle Side Length
Relationships
Math On the Spot
You saw in the Explore Activity that you cannot always form a triangle from
three given line segments.
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Triangle Inequality
The sum of the lengths of
any two sides of a triangle
is greater than the length
of the third side.
4
5
7
Can form a
triangle
4
2
7
Cannot form a
triangle
EXAMPLE 1
6.8.A
Tell whether a triangle can have sides with the given lengths.
Animated
Math
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A 11 cm, 6 cm, 13 cm
STEP 1
Find the sum of the lengths of each pair of sides.
?
?
?
11 + 6 > 13 6 + 13 > 11 11 + 13 > 6
STEP 2
Compare the sum to the third side.
17 > 13 ✓
19 > 11 ✓
24 > 6 ✓
The sum of any two of the given lengths is
greater than the third length.
So, a triangle can have these side lengths.
424
Unit 5
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You can use this relationship to determine if given side lengths can form a triangle.
B 5 ft, 15 ft, 9 ft
STEP 1
Find the sum of the lengths of each pair of sides.
?
?
?
15 + 9 > 5 5 + 9 > 15
5 + 15 > 9
STEP 2
Compare the sum to the third side.
20 > 9 ✓
24 > 5 ✓
Math Talk
14 ≯ 15
Mathematical Processes
Explain why a triangle
with sides measuring 5 in.,
5 in., and 1 foot cannot
be constructed.
The sum of any two of the given lengths is not greater
than the third length.
So, a triangle cannot have these side lengths.
YOUR TURN
Tell whether a triangle can have sides with the given lengths. Explain.
2. 3 cm, 6 cm, 9 cm
3.
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4 m, 5 m, 8 m
Online Assessment
and Intervention
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Using Inequalities to Represent
the Relationship Between Triangle
Side Lengths
Math On the Spot
You can use what you know about the relationship among the lengths of the
sides of a triangle to write an inequality. Then you can use the inequality to
determine if a given value can be the length of an unknown side.
EXAMPL 2
EXAMPLE
6.8.A
Which value could be the length of x?
x = 15
x = 10
4
9
STEP 1
4+9>x
4+9>x
x
Write an inequality.
STEP 2
?
4 + 9 > 15
?
4 + 9 > 10
Substitute each value for x.
STEP 3
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Math Talk
Mathematical Processes
Explain how you know
that the Triangle Inequality
relationship is true for every
equilateral triangle.
Compare the sum to the
given value of x.
The value that could be the length of x is x = 10.
13 ≯ 15
13 > 10 ✓
Lesson 15.1
425
YOUR TURN
Personal
Math Trainer
4. Which value could be the length of x?
Online Assessment
and Intervention
20
x = 35
13
x = 13
x
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Guided Practice
Determine whether a triangle can have sides with the given lengths.
Explain. (Explore Activity and Example 1)
1. 3 cm, 10 cm, 8 cm
2. 10 ft, 10 ft, 18 ft
3. 30 in., 20 in., 40 in.
4. 16 cm, 12 cm, 3 cm
x = 29
x = 45
17
x
22
?
?
ESSENTIAL QUESTION CHECK-IN
6. Explain how you can determine whether three metal rods can be
joined to form a triangle.
426
Unit 5
© Houghton Mifflin Harcourt Publishing Company
5. Which value could be the length of x?
(Example 2)
Name
Class
Date
15.1 Independent Practice
Personal
Math Trainer
6.8.A
7. A map of a new dog park shows that it is
triangular and that the sides measure
18.5 m, 36.9 m, and 16.9 m. Are the
dimensions correct? Explain your reasoning.
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Online
Assessment and
Intervention
10. Geography The map shows the distance
in air miles from Houston to both Austin
and San Antonio.
Austin
146.43 mi
Houston
San Antonio
8. Choose a real world object that you can
cut into three different lengths to form a
triangle. Find three side lengths that form a
triangle and three lengths that do not form
a triangle. For each triangle, give the side
lengths and explain why those lengths do
or do not form a triangle.
189.34 mi
a. What is the greatest possible distance
from Austin to San Antonio?
b. How did you find the answer?
Triangle 1:
© Houghton Mifflin Harcourt Publishing Company
c. What is the least possible distance
from Austin to San Antonio?
Triangle 2:
d. How did you find the answer?
9. Could the three sides of a triangular
shopping mall measure _12 mi, _13 mi, and
_1 mi? Show how you found your answer.
4
Lesson 15.1
427
11. Critical Thinking Two sides of an isosceles triangle measure 3 inches
and 13 inches respectively. Find the length of the third side. Explain
your reasoning.
FOCUS ON HIGHER ORDER THINKING
Work Area
12. Critique Reasoning While on a car trip with her family, Erin saw a sign that
read, “Amarillo 100 miles, Lubbock 80 miles.” She concluded that the distance
from Amarillo to Lubbock is 100 - 80 = 20 miles. Was she right? Explain.
14. Persevere in Problem Solving A metalworker cut an 8-foot length of
pipe into three pieces and welded them to form a triangle. Each of the
3 sections measured a whole number of feet in length. How long was
each section? Explain your reasoning.
428
Unit 5
© Houghton Mifflin Harcourt Publishing Company
13. Make a Conjecture Is there a value of n for which there could be a
triangle with sides of length n, 2n, and 3n? Explain.
LESSON
15.2
?
Sum of Angle
Measures in a
Triangle
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—6.8.A
Extend previous knowledge of
triangles and their properties
to include the sum of angles
in a triangle …
How do you use the sum of angles in a triangle to find an
unknown angle measure?
EXPLORE ACTIVITY
6.8.A
Exploring Angles in a Triangle
Recall that a triangle is a closed figure with three line
segments and three angles. The measures of the angles of a
triangle have a special relationship with one another.
2
A Use a straightedge to draw a large triangle. Label the
angles 1, 2, and 3.
3
1
B Use scissors to cut out the triangle.
C Tear off the three angles. Arrange them
around a point on a line as shown.
1
2
3
D What is the measure of the straight
angle formed by the three angles?
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E What is the sum of the measures of the three angles? Explain.
F Compare your results with those of your classmates. What guess can
you make?
Reflect
1. Justify Reasoning How can you show that your guess is correct?
Lesson 15.2
429
Finding an Angle Measure in
a Triangle
Math On the Spot
Sum of Angle Measures of a Triangle
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The sum of the measures of the
angles in a triangle is 180°.
2
1
3
m∠1 + m∠2 + m∠3 = 180°
EXAMPLE 1
6.8.A
Fountain Place, shown to the right, is a 720-foot Dallas skyscraper. Find the
measure of the unknown angle in the triangle at the top of the building.
The sum of the angle measures
m∠1 + m∠2 + m∠3 = 180° in a triangle is 180°.
x
65° + 65° + x = 180°
Write an equation.
130° + x = 180°
−130°
65°
Add.
−130°
65°
Subtract 130° from both
sides.
x = 50°
The angle at the top of the triangle measures 50°.
Math Talk
Mathematical Processes
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Can a triangle have two
obtuse angles? Why
or why not?
YOUR TURN
Find the unknown angle measures.
2.
D
100°
x
E
K
3.
x
55°
F
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Math Trainer
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and Intervention
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430
Unit 5
J
x=
x=
71°
56°
L
Finding Angles in an
Equilateral Triangle
Recall that an equilateral triangle has three congruent sides and three
congruent angles.
Math On the Spot
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EXAMPL 2
EXAMPLE
6.8.A
Find the angle measures in the equilateral triangle.
3x = 180°
Write an equation.
3x = ____
180°
___
Divide both sides by 3.
3
3
x = 60°
x
x
x
Each angle in an equilateral triangle measures 60°.
Reflect
4. Multiple Representations Write a different equation to find the angle
measures in Example 2. Will the answer be the same? Explain.
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5. Draw Conclusions Triangle ABC is a right triangle. What conclusions
can you draw about the measures of the angles of the triangle?
YOUR TURN
Write an equation to find the unknown angle measure in each triangle.
6. The measures of two of the angles are 25° and 65°.
7. The measures of two of the angles are 60°.
8. The measures of two of the angles are 35°.
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Math Trainer
Online Assessment
and Intervention
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Lesson 15.2
431
Guided Practice
1. The sum of the angle measures in a triangle is
(Explore Activity)
.
Find the unknown angle measure in each triangle. (Examples 1 and 2)
2. m∠R + m∠S + m∠T =
+
S
105°
+x=
T
x
+x=
-
42°
-
R
x=
K
3.
A
x
4.
96°
x
42°
L
28°
C
x=
x=
5. G
M
6.
61°
33°
F
x
28°
H
x=
P x
8. The measures of two of the angles are 50° and 30°.
ESSENTIAL QUESTION CHECK-IN
9. Arlen knows the measures of two angles of a triangle. Explain how he
can find the measure of the third angle. Why does your method work?
432
Unit 5
59° N
x=
7. The measures of two of the angles are 45°.
?
?
B
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J
Name
Class
Date
15.2 Independent Practice
Personal
Math Trainer
6.8.A
Figure ABCD represents
a garden crossed by
___
straight walkway AC. Use the figure for 10–15.
A
16. An observer at point O sees airplane
P directly over airport A. The observer
measures the angle of the plane at 40.5°.
B
P
57°
100°
D
Online
Assessment and
Intervention
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32°
88°
C
O
10. Find m∠DAC.
40.5°
A
Find m∠P.
11. Explain how you found m∠DAC.
The map shows the intersection of three
streets in San Antonio’s River Walk district.
Use the map for 17–18.
B
12. Find m∠BAC.
N
ll
S
48°
© Houghton Mifflin Harcourt Publishing Company
13. Explain how you found m∠BAC.
A
h
y’
n
C
17. Find the measures of the three angles of
the triangle.
14. Find m∠DAB.
15. Explain how you found m∠DAB.
18. Explain how you found the angle
measures.
Lesson 15.2
433
FOCUS ON HIGHER ORDER THINKING
Work Area
19. Persevere in Problem Solving Find the measure of ∠ACB. Explain how
you found your answer.
A
83°
x
D
148°
B
C
20. Communicate Mathematical Ideas Explain how you can use the figure
to find the sum of the measures of the angles of quadrilateral ABCD.
What is the sum?
B
A
C
21. Draw Conclusions Recall that a right triangle is a triangle with one right
angle. One angle of a triangle measures 89.99 degrees. Can the triangle
be a right triangle? Explain your reasoning.
434
Unit 5
© Houghton Mifflin Harcourt Publishing Company
D
LESSON
15.3
?
Relationships
Between Sides and
Angles in a Triangle
ESSENTIAL QUESTION
EXPLORE ACTIVITY
Expressions,
equations, and
relationships—6.8.A
Extend previous knowledge
of triangles and their properties
to include…the relationship
between the lengths of sides
and measures of angles
in a triangle…
How can you use the relationships between side lengths
and angle measures in a triangle to solve problems?
6.8.A
Exploring the Relationship Between
Sides and Angles in a Triangle
There is a special relationship between the lengths of sides
and the measures of angles in a triangle.
A Use geometry software to make triangle ABC.
Make ∠A the smallest angle.
B
C
A
B Choose one vertex and drag it so that you
lengthen the side of the triangle opposite angle
A. Describe what happens to ∠A.
B
C
© Houghton Mifflin Harcourt Publishing Company
A
C Drag the vertex to shorten the side opposite ∠B.
What happens to ∠B?
D Make several new triangles. In each case, note
the locations of the longest and shortest sides
in relation to the largest and smallest angles.
Describe your results.
B
A
C
Lesson 15.3
435
Using the Relationship Between Sides
and Angles in a Triangle
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EXAMPLE 1
My Notes
6.8.A
A Triangle ABC has side lengths of 7 cm, 9 cm,
and 4.5 cm. Use the relationship between the
sides and angles of a triangle to match each
side with its correct length.
B
100°
A
50°
30°
AC = 9 cm
The longest side is opposite the largest angle.
AB = 4.5 cm
The shortest side is opposite the smallest angle.
BC = 7 cm
The midsize side is opposite the midsize angle.
B Triangle ABC has angles measuring 60°, 80°, and
40°. Use the relationship between the sides and
angles of a triangle to match each angle with its
correct measure.
B
20
13
m∠C = 40°
A
The largest angle is opposite the
17.6
longest side.
The smallest angle is opposite the shortest side.
m∠B = 60°
The midsize angle is opposite the midsize side.
m∠A = 80°
YOUR TURN
1.
2.
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and Intervention
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436
Unit 5
AC =
BC =
A
88°
57°
35°
m∠B =
6
m∠C =
A
C
C
Triangle ABC has angle measures of 45°, 58°, and 77°.
Match each angle with its correct measure.
m∠A =
C
B
Triangle ABC has side lengths of 11, 16, and 19.
Match each side with its correct length.
AB =
C
5
7
B
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Math On the Spot
You have seen that in a triangle the largest angle is opposite the longest side
and the smallest angle is opposite the shortest side. It follows that the midsize
angle is opposite the midsize side.
Solving Problems Using
Triangle Relationships
Recall that triangles can be classified by the lengths of their sides. A scalene
triangle has no congruent sides. An isosceles triangle has two congruent sides.
An equilateral triangle has three congruent sides.
EXAMPL 2
EXAMPLE
Problem
Solving
FPO
Math On the Spot
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6.8.A
Brandy is making a quilt. Each block of the quilt is made up of four
triangles. Each triangle is in the shape of a right isosceles triangle.
Two of the side measures of one triangle are 6.4 inches and 9 inches.
Brandy wants to add a ribbon border around one of the triangles. How
much ribbon will she need?
Analyze Information
Rewrite the question as a statement.
• Find the amount of ribbon Brandy will need for a border around one
triangle.
Identify the important information.
• Each quilt piece has the shape of a right isosceles triangle.
• Two sides of the triangle measure 6.4 inches and 9 inches.
Formulate a Plan
You can draw a model and label it with the important information to find
the total length of ribbon that Brandy needs for one triangle.
© Houghton Mifflin Harcourt Publishing Company
Justify and Evaluate
Solve
Think: A right triangle will have one 90° angle.
Since the sum of the angles is 180°, the other
two angles will be congruent and will have a
combined measure of 90°.
90° ÷ 2 = 45°
Label the new information on the model.
90° is the greatest angle measure, so the side
opposite the 90° angle will be the longest side.
The other two angles are congruent, so the sides
opposite those angles are congruent.
The shortest side lengths are 6.4 inches and
6.4 inches. So, Brandy will need
6.4 + 6.4 + 9 = 21.8 inches of ribbon.
45°
90°
45°
90°
45°
longest side
45°
Justify and Evaluate
The solution is reasonable because the quilt piece is in the shape of an
isosceles right triangle and it has two sides measuring 6.4 inches and 9 inches.
Lesson 15.3
437
YOUR TURN
3.
Personal
Math Trainer
Online Assessment
and Intervention
A fence around a rock garden is in the shape of a right triangle. Two
angles measure 30° and 60°. Two sides measure 10 feet and 17.3 feet.
The total length of the fence is 47.3 feet. How long is the side opposite
the right angle?
my.hrw.com
Guided Practice
B
1. Triangle ABC has side lengths of 17, 13, and 24. Match
each side with its correct length. (Example 1)
= 24
= 13
= 17
2. The figure represents a traffic island that has angles
measuring 60°, 20°, and 100°. Match each angle with
its correct measure. (Example 1)
m∠
= 100° m∠
= 20° m∠
105°
A
= 60°
3. Vocabulary Explain how the relationship between
the sides and angles of a triangle applies to equilateral
triangles. (Example 2)
43°
32°
C
N
58 in.
20 in.
M
51 in.
P
?
?
ESSENTIAL QUESTION CHECK-IN
5. Describe the relationship between the lengths of the sides and the
measures of the angles in a triangle.
438
Unit 5
© Houghton Mifflin Harcourt Publishing Company
4. Ramone is building a fence around a vegetable garden in his backyard.
The fence will be in the shape of a right isosceles triangle. Two of the side
measures are 12 feet and 16 feet. Use a problem solving model to find
the total length of fencing he needs. Explain. (Example 2)
Name
Class
Date
15.3 Independent Practice
Personal
Math Trainer
6.8.A
my.hrw.com
Use the figure for 6–8.
Online
Assessment and
Intervention
The figure shows the angle measurements
formed by two fenced-in animal pens that
share a side. Use the figure for 9–10.
G
B
A
57°
58°
58°
F
58.5°
61°
H
6. Critique Reasoning Dustin says that
△FGH is an equilateral triangle because
the sides appear to be the same length. Is
his reasoning valid? Explain.
68°
54°
D
65°
C
_
9. Caitlin says that AC is the longest segment
of fencing because it is opposite 68°, the
largest angle measure in the figure. Is her
reasoning valid? Explain.
© Houghton Mifflin Harcourt Publishing Company
7. What additional information do you need
to know before you can determine which
side of the triangle is the longest? How can
you find it?
10. What is the longest segment of fencing in
△ABC? Explain your reasoning.
8. Which side of the triangle is the longest?
Explain how you found the answer.
11. Find the longest segment of fencing in the
figure. Explain your reasoning.
Lesson 15.3
439
_
_
_
_
12. In triangle ABC, AB is longer than BC and BC is longer than AC.
a. Draw a sketch of triangle ABC.
b. Name the smallest angle in the triangle. Explain your reasoning.
FOCUS ON HIGHER ORDER THINKING
Work Area
Z
13. Persevere in Problem Solving
Determine the shortest line segment in the
figure. Explain how you found the answer.
30°
30°
X
14. Communicate Mathematical Ideas Explain how the relationship
between the sides and angles of a triangle applies to isosceles triangles.
15. Critical Thinking Can a scalene triangle contain a pair of congruent
angles? Explain.
440
Unit 5
W
© Houghton Mifflin Harcourt Publishing Company
Y
MODULE QUIZ
Ready
Personal
Math Trainer
15.1 Determining When Three Lengths Form a Triangle
Online Assessment
and Intervention
Determine whether the three side lengths form a triangle.
my.hrw.com
1. 3, 5, 7
2. 9, 15, 4
3. 17, 5, 23
4. 28, 16, 38
15.2 Sum of Angle Measures in a Triangle
Find the unknown angle measures.
5.
6.
88°
41°
38°
112°
15.3 Relationships Between Sides and Angles in a Triangle
Match each of the given measures to the correct side or angle.
7. 11, 7.5, 13
8. 24°, 44°, 112°
A
89°
16
© Houghton Mifflin Harcourt Publishing Company
D
35°
C
7
12
56°
B
E
F
ESSENTIAL QUESTION
9. How can you describe the relationships among angles and sides in
a triangle?
Module 15
441
Personal
Math Trainer
MODULE 15 MIXED REVIEW
Texas Test Prep
Selected Response
1. The two longer sides of a triangle measure
16 and 22. Which of the following is a
possible length of the shortest side?
A 4
C
B 6
D 19
my.hrw.com
5. Which of these could be the value of x in
the triangle below?
A
50°
11
29
2. Part of a large metal sculpture will be a
triangle formed by welding three bars
together. The artist has four bars that
measure 12 feet, 7 feet, 5 feet, and 3 feet.
Which bar could not be used with two of
the others to form a triangle?
4x
43°
C
87°
22
B
A 5
B 6
A the 3-foot bar
C
B the 5-foot bar
C
Online
Assessment and
Intervention
7
D 10
the 7-foot bar
D the 12-foot bar
3. What is the measure of the missing angle
in the triangle below?
Gridded Response
6. Find m∠Z.
X
F
133°
29°
Y
.
D
65°
56°
A 39°
C
B 49°
D 69°
E
59°
4. The measure of ∠A in △ABC is 88°. The
measure of ∠B is 60% of the measure of
∠A. What is the measure of ∠C?
442
A 39.2°
C
B 52.8°
D 127.2°
Unit 5
91°
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
© Houghton Mifflin Harcourt Publishing Company
Z
Area and Volume
Equations
?
MODULE
16
LESSON 16.1
ESSENTIAL QUESTION
Area of
Quadrilaterals
How can you use area
and volume equations
to solve real-world
problems?
6.8.B, 6.8.D
LESSON 16.2
Area of Triangles
6.8.B, 6.8.D
LESSON 16.3
Solving Area
Equations
6.8.C, 6.8.D
LESSON 16.4
Solving Volume
Equations
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Getty Royalty
Free
6.8.C, 6.8.D
Real-World Video
my.hrw.com
my.hrw.com
Quilting, painting, and other art forms use familiar
geometric shapes, such as triangles and rectangles.
To buy enough supplies for a project, you need to find
or estimate the areas of each shape in the project.
my.hrw.com
Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
443
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this chapter.
my.hrw.com
Use of Parentheses
EXAMPLE
Online
Assessment and
Intervention
1
_
(14) (12 + 18) = _12 (14) (30) Perform operations inside parentheses first.
2
= 7 (30)
Multiply left to right.
= 210
Multiply again.
Evaluate.
1. _12 (3) (5 + 7)
2. _12 (15) (13 + 17)
3. _12 (10) (9.4 + 3.6) 4. _12 (2.1) (3.5 + 5.7)
Area of Square, Rectangles, Triangle
EXAMPLE
Find the area of the
rectangle.
8 ft
3 ft
A = bh
Use the formula for area of a rectangle.
= 8 · 3 Substitute for base and height.
= 24
Multiply.
Area equals 24 square feet.
Find the area of each figure.
6. a square with sides of 7.6 m
7. a rectangle with length 3_41 ft and width 2_21 ft
8. a triangle with base 8.2 cm and height 5.1 cm
444
Unit 5
© Houghton Mifflin Harcourt Publishing Company
5. a triangle with base 6 in. and height 3 in.
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the graphic. You will put one
word in each oval.
Types of Triangles
has three
congruent
sides and three
congruent
angles
contains a 90°
angle
has two equal
sides and two
equal angles
Vocabulary
Review Words
base (base)
✔ equilateral triangle
(triángulo equilátero)
✔ isosceles triangle
(triángulo isósceles)
legs (catetos)
quadrilateral (cuadrilátero)
rectangular prism (prisma
rectangular)
✔ right triangle (triángulo
rectángulo)
volume (volumen)
Preview Words
parallelogram
(paralelogramo)
rhombus (rombo)
trapezoid (trapecio)
Understand Vocabulary
© Houghton Mifflin Harcourt Publishing Company
Match the term on the left to the correct expression on the right.
1. parallelogram
A. A quadrilateral in which all sides are congruent
and opposite sides are parallel.
2. trapezoid
B. A quadrilateral in which opposite sides are
parallel and congruent.
3. rhombus
C. A quadrilateral in which two sides are parallel.
Active Reading
Booklet Before beginning the module, create
a booklet to help you learn the concepts in this
module. Write the main idea of each lesson on
each page of the booklet. As you study each
lesson, write important details that support the
main idea, such as vocabulary and formulas.
Refer to your finished booklet as you work on
assignments and study for tests.
Module 16
445
MODULE 16
Unpacking the TEKS
Understanding the TEKS and the vocabulary terms in the TEKS
will help you know exactly what you are expected to learn in this
module.
6.8.C
Write equations that represent
problems related to the area
of rectangles, parallelograms,
trapezoids, and triangles and
volume of right rectangular
prisms where dimensions are
positive rational numbers.
What It Means to You
You will use the formula for the area of a figure to write an
equation that can be used to solve a problem.
UNPACKING EXAMPLE 6.8.C
The Hudson Middle School wrestling team won the state
tournament and was awarded a triangular pennant to display in
the school gymnasium. The pennant has an area of 2.25 square
meters. The base of the pennant is 1.5 meters long. Write an
equation to find the height of the pennant.
A = _12 bh
2.25 = _12 (1.5)h
2.25 = 0.75h
1.5 m
6.8.D
Determine solutions for
problems involving the area
of rectangles, parallelograms,
trapezoids, and triangles and
volume of right rectangular
prisms where dimensions are
positive rational numbers.
What It Means to You
You will use the formula for the volume of a rectangular prism.
UNPACKING EXAMPLE 6.8.D
Jala has an aquarium in the shape of a rectangular prism with
a volume of 2,160 cubic inches. The length is 15 inches and the
width is 12 inches. Find the height of the aquarium.
v=l·w·h
2,160 = 15 · 12 · h
2,160 = 180 · h
2,160
_____
=h
180
Visit my.hrw.com
to see all
the
unpacked.
my.hrw.com
446
Unit 5
12 = h
The height of the aquarium is 12 inches.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: GK Hart/Vikki
Hart/Getty Images
An equation to find the height of the pennant is 2.25 = 0.75h.
LESSON
16.1
?
Area of
Quadrilaterals
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—
6.8.B Model area
formulas for parallelograms,
trapezoids ... by decomposing
and rearranging parts of
these shapes. Also 6.8.D
How can you find the areas of parallelograms,
rhombuses, and trapezoids?
EXPLORE ACTIVITY
6.8.B
Area of a Parallelogram
Recall that a rectangle is a special type of parallelogram.
A Draw a large parallelogram on grid paper. Cut out your parallelogram.
B Cut your parallelogram on the dashed line as shown. Then move
the triangular piece to the other side of the parallelogram.
height (h)
base (b)
width (w)
length (l)
C What figure have you formed?
Does this figure have the same area as the parallelogram?
© Houghton Mifflin Harcourt Publishing Company
base of parallelogram =
of rectangle
height of parallelogram =
area of parallelogram =
of rectangle
Math Talk
of rectangle
Mathematical Processes
What is the formula for the area of this figure? A =
or
D What is the formula for the area of a parallelogram? A =
How is the relationship
between the length and width
of a rectangle similar to the
relationship between the
base and height of a
parallelogram?
Area of a Parallelogram
The area A of a parallelogram is the
product of its base b and its height h.
A = bh
h
b
Lesson 16.1
447
EXPLORE ACTIVITY (cont’d)
Reflect
1. Find the area of the parallelogram.
7 cm
A=
14 cm
Finding the Area of a Trapezoid
Math On the Spot
To find the formula for the area of a trapezoid, notice that two copies of the
same trapezoid fit together to form a parallelogram. Therefore, the area of the
trapezoid is _12 the area of the parallelogram.
my.hrw.com
h
b
A=
The height of the parallelogram is the same as the height
of the trapezoid. The base of the parallelogram is the sum
of the two bases of the trapezoid.
b
↓
·h
A = (b1 + b2) · h
Area of a Trapezoid
b1
h
b2
EXAMPLE 1
Math Talk
A section of a deck is in the shape of a trapezoid. What is the area of this
section of the deck?
Mathematical Processes
Does it matter which of
the trapezoid’s bases is
substituted for b1 and which
is substituted for b2?
Why or why not?
Animated
Math
my.hrw.com
448
Unit 5
6.8.D
17 ft
b1 = 17
b2 = 39
h = 16
Use the formula for area of a trapezoid.
1 h(b + b )
A = __
2
2 1
1 · 16(17 + 39)
= __
2
1 · 16(56)
= __
2
16 ft
39 ft
Substitute.
Add inside the parentheses.
= 8 · 56
1 and 16.
Multiply __
2
= 448 square feet
Multiply.
© Houghton Mifflin Harcourt Publishing Company
The area of a trapezoid is half its height
multiplied by the sum of the lengths
of its two bases.
1 h(b + b )
A = __
2
2 1
YOUR TURN
2. Another section of the deck is also shaped like a trapezoid. For this
section, the length of one base is 27 feet, and the length of the other
base is 34 feet. The height is 12 feet. What is the area of this section of
the deck? A =
f t2
Personal
Math Trainer
Online Assessment
and Intervention
my.hrw.com
Finding the Area of a Rhombus
A rhombus is a quadrilateral in which all sides are congruent and opposite
sides are parallel. A rhombus can be divided into four triangles that can then
be rearranged into a rectangle.
Math On the Spot
my.hrw.com
h
b
A=b · h
↓
↓
The base of the rectangle is the same length as one of
the diagonals of the rhombus. The height of the rectangle
is _21 the length of the other diagonal.
A = d1 · _12 d2
Area of a Rhombus
The area of a rhombus is half of the
product of its two diagonals.
© Houghton Mifflin Harcourt Publishing Company
A = _12 d1d2
d2
d1
EXAMPL 2
EXAMPLE
6.8.B
Cedric is constructing a kite in the shape of a rhombus.
The spars of the kite measure 15 inches and 24 inches.
How much fabric will Cedric need for the kite?
To determine the amount of fabric needed,
find the area of the kite.
d1 = 15
15 in.
d2 = 24
Use the formula for area of a rhombus.
A = _12 d1d2
24 in.
= _12 (15)(24)
Substitute.
= 180 square inches
Multiply.
Lesson 16.1
449
YOUR TURN
Find the area of each rhombus.
Personal
Math Trainer
3. d1 = 35 m; d2 = 12 m
Online Assessment
and Intervention
A=
4. d1 = 9.5 in.; d2 = 14 in.
A=
m2
in2
my.hrw.com
5. d1 = 10 m; d2 = 18 m
A=
6. d1 = 8_14 ft; d2 = 40 ft
A=
m2
ft2
Guided Practice
1. Find the area of the parallelogram. (Explore Activity)
A = bh
=(
9 in.
)(
=
)
13 in.
in.2
2. Find the area of the trapezoid. (Example 1)
A = _12h(b1 + b2)
= _12
(
=
)(
+ 9 cm
)
14 cm
cm2
15 cm
A = _12 d1d2
= _12
(
=
?
?
)(
)
18 in.
11 in.
in.2
ESSENTIAL QUESTION CHECK-IN
4. How can you find the areas of parallelograms, rhombuses, and trapezoids?
450
Unit 5
© Houghton Mifflin Harcourt Publishing Company
3. Find the area of the rhombus. (Example 2)
Name
Class
Date
16.1 Independent Practice
Personal
Math Trainer
6.8.B, 6.8.D
5. Rearrange the parts of the parallelogram
to form a rectangle. Find the area of the
parallelogram and the area of the rectangle.
What is the relationship between the areas?
my.hrw.com
Online
Assessment and
Intervention
9. The seat of a bench is in the shape of a
trapezoid with bases of 6 feet and 5 feet
and a height of 1.5 feet. What is the area of
the seat?
6 cm
10. A kite in the shape of a rhombus has
diagonals that are 25 inches long and 15
inches long. What is the area of the kite?
14 cm
6. What is the area of a parallelogram that has
a base of 12_34 in. and a height of 2_12 in.?
7. Draw a copy of the trapezoid to form a
parallelogram. Find the area of the trapezoid
and the area of the parallelogram. What is
the relationship between the areas?
© Houghton Mifflin Harcourt Publishing Company
42 in.
11. A window in the shape of a parallelogram
has a base of 36 inches and a height of 45
inches. What is the area of the window?
12. Communicate Mathematical Ideas Find
the area of the figure. Explain how you
found your answer.
10 ft
6 ft
24 in.
12 ft
36 in.
18 ft
8. The bases of a trapezoid are 11 meters and
14 meters. Its height is 10 meters. What is
the area of the trapezoid?
Lesson 16.1
451
13. Multistep A parking space shaped like a parallelogram has a base of
17 feet and a height is 9 feet. A car parked in the space is 16 feet long and
6 feet wide. How much of the parking space is not covered by the car?
FOCUS ON HIGHER ORDER THINKING
Work Area
14. Critique Reasoning Simon says that to find the area of a trapezoid, you
can multiply the height by the top base and the height by the bottom
base. Then add the two products together and divide the sum by 2. Is
Simon correct? Explain your answer.
16. Critique Reasoning Find the area
of the trapezoid using the formula
A = _12 h(b1 + b2). Decompose the
trapezoid into a rectangle and a
triangle and find the area of each. Then
find the sum of the two areas. Compare
this sum with the area of the trapezoid.
452
Unit 5
12 cm
8 cm
6 cm
© Houghton Mifflin Harcourt Publishing Company
15. Multistep The height of a trapezoid is 8 in. and its area is 96 in2. One
base of the trapezoid is 6 inches longer than the other base. What are the
lengths of the bases? Explain how you found your answer.
LESSON
16.2 Area of Triangles
?
Expressions,
equations, and
relationships—
6.8.B Model area
formulas for ... triangles by
decomposing and rearranging
parts of these shapes. Also
6.8.D
ESSENTIAL QUESTION
How do you find the area of a triangle?
EXPLORE ACTIVITY 1
6.8.B
h
Area of a Right Triangle
b
A Draw a large rectangle on grid paper.
What is the formula for the area of a rectangle? A =
B Draw one diagonal of your rectangle.
The diagonal divides the rectangle into
Each one represents
.
of the rectangle.
Use this information and the formula for area of a rectangle to
write a formula for the area of a right triangle. A =
Reflect
© Houghton Mifflin Harcourt Publishing Company
1. Communicate Mathematical Ideas In the formula for the area
of a right triangle, what do b and h represent?
EXPLORE ACTIVITY 2
6.8.B
Area of a Triangle
A Draw a large triangle on grid paper. Do not draw a right triangle.
B Cut out your triangle. Then trace around it to make a copy of
your triangle. Cut out the copy.
C Cut one of your triangles into two pieces by cutting through
one angle directly across to the opposite side. Now you have
three triangles — one large triangle and two smaller triangles.
Lesson 16.2
453
EXPLORE ACTIVITY 2 (cont’d)
When added together, the areas of the two smaller triangles
equal the
of the large triangle.
D Arrange the three triangles into a rectangle.
What fraction of the rectangle does the large
h
triangle represent?
The area of the rectangle is A = bh. What is the area
b
of the large triangle? A =
How does this formula compare to the formula for the area of
a right triangle that you found in Explore Activity 1?
Reflect
2. Communicate Mathematical Ideas What type of angle is formed by
the base and height of a triangle?
Finding the Area of a Triangle
Math On the Spot
The area A of a triangle is half the product
of its base b and its height h.
my.hrw.com
h
A = _12 bh
b
EXAMPLE 1
6.8.D
Find the area of each triangle.
b = 20 meters
A
8m
20 m
454
Unit 5
h = 8 meters
A = _12 bh
= _12 (20 meters) (8 meters)
Substitute.
= 80 square meters
Multiply.
© Houghton Mifflin Harcourt Publishing Company
Area of a Triangle
Find the area of each triangle.
b = 12 inches
B
h = 5 inches
A = _12 bh
5 in.
= _12 (12 inches) (5 inches) Substitute.
12 in.
= 30 square inches
Multiply.
YOUR TURN
Find the area of the triangle.
Personal
Math Trainer
3.
Online Assessment
and Intervention
14 in.
8.5 in.
my.hrw.com
Math Talk
A=
Mathematical Processes
Why can you also write the
formula for the area of a
bh
triangle as A = __
?
2
Problem Solving Using Area
of Triangles
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©JOHN
MACDOUGALL/AFP/Getty Images
You can use the formula for the area of a triangle to solve real-world problems.
Math On the Spot
EXAMPL 2
EXAMPLE
6.8.D
Each triangular face of the Pyramid of Peace in
Kazakhstan is made up of 25 smaller equilateral
triangles. These triangles have measurements as
shown in the diagram. What is the area of one of the
smaller equilateral triangles?
STEP 1
Identify the length of the base and the
height of the triangle.
my.hrw.com
10.4 m
12 m
b = 12 m and h = 10.4 m
STEP 2
Use the formula to find the area of
the triangle.
A = _12 bh
= _12 (12) (10.4)
Substitute.
Multiply.
= 62.4
The area of one small equilateral triangle is 62.4 m2.
Lesson 16.2
455
YOUR TURN
4.
Personal
Math Trainer
Amy needs to order a shade for a triangular-shaped window that has
a base of 6 feet and a height of 4 feet. What is the area of the shade?
Online Assessment
and Intervention
my.hrw.com
Guided Practice
1. Show how you can use a copy of the triangle to form a rectangle. Find the
area of the triangle and the area of the rectangle. What is the relationship
between the areas? (Explore Activities 1 and 2, Example 1)
8 in.
14 in.
Triangle: A = _12 bh
Rectangle: A = bh
=
)(
)
in2
=(
)(
=
in2
2. A pennant in the shape of a triangle has a base of
12 inches and a height of 30 inches. What is the area
of the pennant? (Example 2)
A = _12 bh
= _12 (
=
?
?
)(
)
in2
ESSENTIAL QUESTION CHECK-IN
3. How do you find the area of a triangle?
456
Unit 5
#
)
'/ 3Ý
/9/ 4 %
© Houghton Mifflin Harcourt Publishing Company
= _12 (
Name
Class
Date
16.2 Independent Practice
Personal
Math Trainer
6.8.B, 6.8.D
my.hrw.com
Online
Assessment and
Intervention
Find the area of each triangle.
4.
5.
20 ft
10 cm
24 ft
15 cm
6.
7.
18 ft
32 ft
17 in.
12 in.
8. What is the area of a triangle that has a
base of 15_14 in. and a height of 18 in.?
© Houghton Mifflin Harcourt Publishing Company
10. A triangular plot of land has the dimensions
shown in the diagram. What is the area of
the land?
9. A right triangle has legs that are 11 in. and
13 in. long. What is the area of the triangle?
11. The front part of a tent has the dimensions
shown in the diagram. What is the area of
this part of the tent?
20 km
5 ft
30 km
12. Multistep The sixth-grade art students are
making a mosaic using tiles in the shape
of right triangles. Each tile has leg measures
of 3 centimeters and 5 centimeters. If there
are 200 tiles in the mosaic, what is the area
of the mosaic?
8 ft
13. Critique Reasoning Monica has a
triangular piece of fabric. The height of the
triangle is 15 inches and the triangle’s base
is 6 inches. Monica says that the area of
2
the fabric is 90 in . What error did Monica
make? Explain your answer.
Lesson 16.2
457
14. Show how you can use the given triangle and its two smaller
right triangles to form a rectangle. What is the relationship
between the area of the original triangle and the area of the
rectangle?
12 in.
18 in.
FOCUS ON HIGHER ORDER THINKING
Work Area
15. Communicate Mathematical Ideas Explain how the areas of a triangle
and a parallelogram with the same base and height are related.
17. What If? A right triangle has an area of 18 square inches.
a. If the triangle is an isosceles triangle, what are the lengths
of the legs of the triangle?
b. If the triangle is not an isosceles triangle, what are all the possible
lengths of the legs, if the lengths are whole numbers?
458
Unit 5
© Houghton Mifflin Harcourt Publishing Company
16. Analyze Relationships A rectangle and a triangle have the same area. If
their bases are the same lengths, how do their heights compare? Justify
your answer.
LESSON
16.3
?
Solving Area
Equations
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—6.8.C
Write equations that represent
problems related to the area
of rectangles, parallelograms,
trapezoids, and triangles …
where dimensions are positive
rational numbers. Also 6.8.D
How do you use equations to solve problems about area of
rectangles, parallelograms, trapezoids, and triangles?
Problem Solving Using the Area
of a Triangle
Recall that the formula for the area of a triangle is A = _12 bh. You can also use the
formula to find missing dimensions if you know the area and one dimension.
Math On the Spot
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EXAMPL 1
EXAMPLE
6.8.D
The Hudson High School wrestling
team just won the state tournament
and has been awarded a triangular
pennant to hang on the wall in the
school gymnasium. The base of the
pennant is 1.5 feet long. It has
an area of 2.25 square feet.
What is the height of the pennant?
© Houghton Mifflin Harcourt Publishing Company
A = _12 bh
1.5 ft
Write the formula.
2.25 = _12 (1.5)h
Use the formula to write an equation.
2.25 = 0.75h
1
Multiply __
and 1.5.
2
2.25 ____
____
= 0.75
h
0.75
0.75
Divide both sides of the equation by 0.75.
3= h
Math Talk
Mathematical Processes
How can you use units in
the formula to confirm that
the units for the height
are in feet?
The height of the pennant is 3 feet.
YOUR TURN
1. Renee is sewing a quilt whose pattern contains right triangles.
Each quilt piece has a height of 6 in. and an area of 24 in2.
How long is the base of each quilt piece?
Personal
Math Trainer
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and Intervention
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Lesson 16.3
459
Writing Equations Using the Area
of a Trapezoid
You can use the formula for area of a trapezoid to write an equation
to solve a problem.
Math On the Spot
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EXAMPLE 2
6.8.C
A garden in the shape of a trapezoid has an area of
44.4 square meters. One base is 4.3 meters and the
other base is 10.5 meters long. The height of the
trapezoid is the width of the garden. How wide is
the garden?
A = _12 h (b1 + b2)
Math Talk
Mathematical Processes
How can you check
that the answer is
reasonable?
4.3 m
10.5 m
Write the formula.
44.4 = _12 h (4.3 + 10.5)
Use the formula to write an equation.
44.4 = _12 h (14.8)
Add inside parentheses.
44.4 = 7.4 h
1 and 14.8.
Multiply __
2
44.4 ___
____
= 7.4
h
7.4
7.4
Divide both sides of the equation by 7.4.
6= h
The garden is 6 meters wide.
Reflect
Communicate Mathematical Ideas Explain why the first step after
substituting is addition.
YOUR TURN
3.
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460
Unit 5
The cross section of a water bin is shaped like a trapezoid. The bases of the
trapezoid are 18 feet and 8 feet long. It has an area of 52 square feet. What
is the height of the cross section?
© Houghton Mifflin Harcourt Publishing Company
2.
Solving Multistep Problems
You can write equations to solve real-world problems using relationships
in geometry.
EXAMPL 3
EXAMPLE
Problem
Solving
6.8.D
Math On the Spot
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John and Mary are using rolls of fabric to make a rectangular stage curtain
for their class play. The rectangular piece of fabric on each roll measures
2.5 feet by 15 feet. If the area of the curtain is 200 square feet, what is the
least number of rolls they need?
Analyze Information
Rewrite the question as a statement.
• Find the least number of rolls of fabric needed to cover an area
of 200 ft2.
List the important information.
• Each roll of fabric is a 2.5 foot by 15 foot rectangle.
• The area of the curtain is 200 square feet.
Formulate a Plan
Write an equation to find the area of each roll of fabric.
Use the area of the curtain and the area of each roll to write an equation
to find the least number of rolls.
Solve
STEP 1
Write an equation to find the area of each roll of fabric.
A = lw
© Houghton Mifflin Harcourt Publishing Company
A = 15 · 2.5
A = 37.5 ft2
STEP 2
Write an equation to find the least number of rolls.
n = 200 ÷ 37.5
n = 5_13
STEP 3
The problem asks for the least number of rolls needed. Since 5 rolls
will not be enough, they will need 6 rolls to make the curtain.
John and Mary will need 6 rolls of fabric to make the curtain.
Justify and Evaluate
The area of each roll is about 38 ft2. Since 38 ft2 · 6 = 228 ft2, the answer
is reasonable.
Lesson 16.3
461
YOUR TURN
Personal
Math Trainer
4.
Online Assessment
and Intervention
A parallelogram-shaped field in a park needs sod. The parallelogram has a
base of 21.5 meters and a height of 18 meters. The sod is sold in pallets of
50 square meters. How many pallets of sod are needed to fill the field?
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Guided Practice
1. A triangular bandana has an area of 70 square inches. The height of the
triangle is 8_34 inches. Write and solve an equation to find the length of the
base of the triangle. (Example 1)
2. The top of a desk is shaped like a trapezoid. The bases of the trapezoid
are 26.5 and 30 centimeters long. The area of the desk is 791 square
centimeters. The height of the trapezoid is the width of the desk.
Write and solve an equation to find the width of the desk. (Example 2)
3. Taylor wants to paint his rectangular deck that is 42 feet long and 28 feet
wide. A gallon of paint covers about 350 square feet. How many gallons
of paint will Taylor need to cover the entire deck? (Example 3)
Write an equation to find the
of the deck.
Write an equation to find the
.
Write and solve the equation.
Taylor will need
?
?
gallons of paint.
ESSENTIAL QUESTION CHECK-IN
4. How do you use equations to solve problems about area of rectangles,
parallelograms, trapezoids, and triangles?
462
Unit 5
© Houghton Mifflin Harcourt Publishing Company
Write and solve the equation.
Name
Class
Date
16.3 Independent Practice
6.8.C, 6.8.D
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Assessment and
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5. A window shaped like a parallelogram has an area of 18 _13 square
feet. The height of the window is 3 _13 feet. How long is the base of
the window?
6. A triangular sail has a base length of 2.5 meters. The area of the sail
is 3.75 square meters. How tall is the sail?
7. A section in a stained glass window is shaped like a trapezoid. The
top base is 4 centimeters and the bottom base is 2.5 centimeters
long. If the area of the section of glass is 3.9 square centimeters,
how tall is the section?
8. Multistep Amelia wants to paint three walls in her family room.
Two walls are 26 feet long by 9 feet wide. The other wall is 18 feet
long by 9 feet wide.
a. What is the total area of the walls that Amelia wants to paint?
b. Each gallon of paint covers about 250 square feet. How many
gallons of paint should Amelia buy to paint the walls?
© Houghton Mifflin Harcourt Publishing Company
9. Critical Thinking The area of a triangular block is 64 square
inches. If the base of the triangle is twice the height, how long are
the base and the height of the triangle?
10. Multistep Alex needs to varnish the top and the bottom of a
dozen rectangular wooden planks. The planks are 8 feet long and
3 feet wide. Each pint of varnish covers about 125 square feet and
costs $3.50.
a. What is the total area that Alex needs to varnish?
b. How much will it cost Alex to varnish all the wooden planks?
11. Multistep Leia cuts congruent triangular patches with an area
of 45 square centimeters from a rectangular piece of fabric that is
18 centimeters long and 10 centimeters wide. How many of the
patches can Leia cut from 32 pieces of the fabric?
12. Multistep A farmer needs to buy fertilizer for two fields. One field is
shaped like a trapezoid, and the other is shaped like a triangle. The
trapezoidal field has bases that are 35 and 48 yards and a height of 26
yards. The triangular field has the same height and a base of 39 yards.
Each bag of fertilizer covers 150 square yards. Use a problem solving
model to find how many bags of fertilizer the farmer needs to buy.
Lesson 16.3
463
13. A tennis court for singles play is 78 feet long and 27 feet wide.
a. The court for doubles play has the same length but is 9 feet
wider than the court for singles play. How much more area is
covered by the tennis court used for doubles play?
b. The junior court for players 8 and under is 36 feet long and 18
feet wide. How much more area is covered by the tennis court
used for singles play than by the junior court?
c. The court for players 10 and under has the same width but is
18 feet shorter than the court for singles play. How much more
area is covered by the tennis court used for singles play?
14. Draw Conclusions The cross section of a metal ingot is a
trapezoid. The cross section has an area of 39 square centimeters.
The top base of the cross section is 12 centimeters. The length of
the bottom base is 2 centimeters greater than the top base. How
tall is the metal ingot? Explain.
Work Area
FOCUS ON HIGHER ORDER THINKING
15. Analyze Relationships A mirror is made
of two congruent parallelograms as shown
in the diagram. The parallelograms have a
combined area of 9 _13 square yards. The height
of each parallelogram is 1 _13 yards.
b. What is the area of the smallest rectangle of wall that the
mirror could fit on?
16. Persevere in Problem Solving
A watercolor painting is 20 inches
long by 9 inches wide. Ramon makes
a mat that adds a 1-inch-wide border
around the painting. What is the area
of the mat?
464
Unit 5
1 in.
20 in.
9 in.
© Houghton Mifflin Harcourt Publishing Company
1 yd
—
2
a. How long is the base of each
parallelogram?
LESSON
16.4
?
Solving Volume
Equations
ESSENTIAL QUESTION
Equations,
expressions, and
relationships—
6.8.C Write equations
that represent problems
related to . . . volume of right
rectangular prisms where
dimensions are positive
rational numbers. Also 6.8.D
How do you write equations to solve problems involving
volume of right rectangular prisms?
Problem Solving by Finding Volume
To find the volume of a box, which is in the shape of a rectangular prism, you
can multiply the length, the width, and the height. The volume of a threedimensional shape is always in cubic units, such as cubic meters or cubic inches.
Math On the Spot
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Volume of a Rectangular Prism
The volume V of a rectangular prism is the product
of its length ℓ, its width w, and its height h.
h
V = ℓwh
Math Talk
ℓ
Mathematical Processes
w
Why are V = ℓwh and V = Bh
both formulas for the
volume of a rectangular
prism?
© Houghton Mifflin Harcourt Publishing Company
EXAMPL 1
EXAMPLE
6.8.D
A rectangular swimming pool is
25 meters long and 17 _12 meters wide.
It has an average depth of 1_12 meters.
What is the volume of the pool?
1
12 m
17 12 m
Label the rectangular prism to represent
the pool.
ℓ = 25 meters
w = 17 _12 meters
25 m
h = 1 _12 meters
Use the formula to write an equation.
V = ℓwh
V = 25 · 17 _12 · 1 _12
35 _
·3
= 25 · __
2 2
2,625
Write mixed numbers as fractions greater than 1.
= ____
4
Multiply.
= 656 _14 cubic meters
Write as a mixed number in simplest form.
Lesson 16.4
465
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
1. Miguel has a toolbox that measures 18 _12 inches by 12 _12 inches
by 4 inches. What is the volume of the toolbox?
V=
cubic inches
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Writing Equations Using the Volume
of a Rectangular Prism
Math On the Spot
You can use the formula for the volume of a rectangular prism to write an
equation. Then solve the equation to find missing measurements for a prism.
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EXAMPLE 2
6.8.C
Samuel has an ant farm with a volume of
375 cubic inches. The width of the
ant farm is 2.5 inches and the length is
15 inches. What is the height of
Samuel’s ant farm?
V = ℓwh
Write the formula.
375 = 15 · 2.5 · h
Use the formula to write
an equation.
Multiply.
375
37.5h
____
= ____
37.5
37.5
Divide both sides of the equation by 37.5.
10 = h
The height of the ant farm is 10 inches.
Reflect
2. Communicate Mathematical Ideas Explain how you would find the
solution to Example 2 using the formula V = Bh.
YOUR TURN
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and Intervention
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466
Unit 5
?
3. Find the height of this shape, which has a
15
volume of __
cubic feet.
16
3
4 ft
1
2 ft
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
Thom Lang/CORBIS
375 = 37.5h
Solving Multistep Problems
One cubic foot of water equals approximately 7.5 gallons and weighs
approximately 62.43 pounds.
EXAMPL 3
EXAMPLE
6.8.D
Math On the Spot
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The classroom aquarium holds 30 gallons of water. It is 0.8 feet wide and
has a height of 2 feet. Find the length of the aquarium.
Divide the total number
STEP 1 Find the volume of the classroom
of gallons by the unit
aquarium in cubic feet.
rate to find the number
30
gallons
of cubic feet.
__________________ = 4 cubic feet
7.5 gallons per cubic foot
The volume of the classroom aquarium is 4 cubic feet.
STEP 2
Find the length of the aquarium.
V = ℓwh
Write the formula for volume.
4 = ℓ · 0.8 · 2
Use the formula to write an equation.
4 = ℓ(1.6)
Multiply.
ℓ(1.6)
4
___
= ____
1.6
1.6
Divide both sides of the equation by 1.6.
2.5 = ℓ
The length of the classroom aquarium is 2.5 feet.
Reflect
© Houghton Mifflin Harcourt Publishing Company
4. Persevere in Problem Solving How much does the water in the
classroom aquarium weigh? Explain.
YOUR TURN
5. An aquarium holds 33.75 gallons of water. It has a length of 2 feet and
a height of 1.5 feet. What is the volume of the aquarium? What is the
width of the aquarium? Explain.
Personal
Math Trainer
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and Intervention
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Lesson 16.4
467
Guided Practice
1. Find the volume of this rectangular prism. (Example 1)
1
1 2 in.
V = ℓwh
V = 3_12 ·
·
V = _____ · ______ · _____ · ______
2
2
1
4
1
3 2 in.
3 in.
V=
The volume of the rectangular prism is
cubic inches.
2. Write an equation to find the width of the rectangular prism. Show your
work. (Example 2)
V = 6,336 cm3
18 cm
? cm
16 cm
?
?
ESSENTIAL QUESTION CHECK-IN
4. How do you solve problems about volume of right rectangular prisms?
468
Unit 5
© Houghton Mifflin Harcourt Publishing Company
3. One red clay brick weighs 5.76 pounds. The brick is 8 inches long and
2_14 inches wide. If the clay weighs 0.08 pounds per cubic inch, what is the
volume of the brick? Write an equation to find the height of the brick.
Show your work. (Example 3)
Name
Class
Date
16.4 Independent Practice
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5. Jala has an aquarium in the shape of a rectangular prism
with the dimensions shown. What is the height of the
aquarium?
Height =
6. Find the volume of a juice box that is 3 in. by 1_21 in. by 4 in.
Online
Assessment and
Intervention
?
V = 3,758.75
cubic inches
12.5 in.
24.25 in.
Volume =
7. Find the width of a cereal box that has a volume of 3,600 cm3
and is 20 cm long and 30 cm high.
Width =
8. Bill has a box of markers that has a base of 8 cm by 20 cm and a height
of 6 cm. Martin’s pencil box has a height of 4 cm and a base that is 15 cm
by 16 cm. Bill says his marker box has the same volume as Martin’s pencil
box. Is Bill right? Explain.
© Houghton Mifflin Harcourt Publishing Company
9. Physical Science A small bar of gold measures 40 mm by 25 mm by 2 mm.
One cubic millimeter of gold weighs about 0.0005 ounces. Find the volume
in cubic millimeters and the weight in ounces of this small bar of gold.
10. History The average stone on the lowest level of the Great Pyramid in
Egypt was a rectangular prism 5 feet long by 5 feet high by 6 feet deep
and weighed 15 tons. What was the volume of the average stone? How
much did one cubic foot of this stone weigh?
11. A freshwater fish is healthiest when there is at least one gallon of water for
every inch of its body length. Roshel wants to put a goldfish that is about
2_12 inches long in her tank. Roshel’s tank is 7 inches long, 5 inches wide, and
7 inches high. The volume of 1 gallon of water is about 231 cubic inches.
a. How many gallons of water would Roshel need for the fish?
b. What is the volume of Roshel’s tank?
c. Is her fish tank large enough for the fish? Explain.
Lesson 16.4
469
12. A box of crackers is a rectangular box with the dimensions
shown. The box is one-fourth full. What is the volume of
4 in.
crackers in the box?
4 in.
FOCUS ON HIGHER ORDER THINKING
8 in.
Work Area
13. Multistep Larry has a clay brick that is 7 inches long, 3.5 inches wide, and
1.75 inches thick, the same size as the gold stored in Ft. Knox in the form
of gold bars. Find the volume of this brick. If the weight of the clay in the
brick is 0.1 pound per cubic inch and the weight of the gold is 0.7 pounds
per cubic inch, find the weight of the brick and the gold bar. Round all
answers the nearest tenth.
Volume of the brick or bar =
Weight of the brick =
Weight of the gold bar =
cubic inches
pounds
pounds
15. Multiple Representations Use the formula V = Bh to write a different
version of this formula that you could use to find the area of the base B of
a rectangular prism if you know the height h and the volume V. Explain
what you did to find this equation.
16. Communicate Mathematical Ideas Explain how you could find the
volume of a cube that has an edge of e.
17. Justify Reasoning Mariel says that a jewelry box that is 3 inches high,
4_12 inches wide, and 5 inches long has a volume of 67_12 inches. Katy says
that answer is not quite correct. What is the error in Mariel’s answer?
470
Unit 5
© Houghton Mifflin Harcourt Publishing Company
14. Represent Real-World Problems Luisa’s toaster oven, which is in the
shape of a rectangular prism, has a base that is 55 cm long by 40 cm
wide. It is 30 cm high. Luisa wants to buy a different oven with the same
volume but a smaller length, so it will fit better on her kitchen counter.
What is a possible set of dimensions for this different oven?
MODULE QUIZ
Ready
Personal
Math Trainer
16.1 Area of Quadrilaterals
1. Find the area of the figure.
Online Assessment
and Intervention
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1 yd
12 —
2
1 yd
17 —
5
16.2 Area of Triangles
2. Find the area of the triangle.
16.3 Solving Area Equations
14 ft
17 ft
3. A triangular pane of glass has a height of 30 inches and an area of
270 square inches. What is the length of the base of the pane?
4. A tabletop in the shape of a trapezoid has an area of 6,550 square
centimeters. Its longer base measures 115 centimeters, and the
shorter base is 85 centimeters. What is the height?
© Houghton Mifflin Harcourt Publishing Company
16.4 Solving Volume Equations
5. A rectangular shoebox has a volume of 728 cubic inches. The base
of the shoebox measures 8 inches by 6.5 inches. How long is the
shoebox?
ESSENTIAL QUESTION
6. How can you use equations to solve problems involving area
and volume?
Module 16
471
Personal
Math Trainer
MODULE 16 MIXED REVIEW
Texas Test Prep
Selected Response
1. What is the area of the rhombus shown
below?
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Online
Assessment and
Intervention
4. The trapezoid below has an area of
1,575 cm2.
63 cm
23 in.
28 in.
27 cm
Which equation could you solve to find the
height of the trapezoid?
A 161 in2
C
B 322 in2
D 966 in2
644 in2
2. What is the area of the triangle shown
below?
A 45h = 1,575
C
850.5h = 1,575
B 90h = 1,575
D 1,701h = 1,575
Gridded Response
5. Cindy is designing a rectangular fountain
in a courtyard. The rest of the courtyard
will be covered in stone.
22 ft
Courtyard
6 ft
3.7 mm
12 ft
Fountain
A 4.44 mm2
B 5.92 mm
C
2
8.88 mm2
The part of the courtyard that will be
covered in stone has an area of 246 ft2.
What is the width of the fountain in feet?
D 17.76 mm2
3. A rectangular prism has a volume of 912
cubic meters. It has a length of 19 meters
and a width of 12 meters. Which equation
could be solved to find the height of the
rectangular prism?
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
A 114h = 912
5
5
5
5
5
5
B 228h = 912
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
C
15.5h = 912
D 31h = 912
472
.
Unit 5
© Houghton Mifflin Harcourt Publishing Company
4.8 mm
UNIT 5
Study Guide
MODULE
MODULE
?
15
15
Review
Angles, Triangles, and
Equations
Key Vocabulary
parallelogram
(paralelogramo)
rhombus (rombo)
trapezoid (trapecio)
ESSENTIAL QUESTION
How can you use angles, triangles, and equations to solve
real-world problems?
EXAMPLE 1
Find the missing angle measure in each triangle.
A
B
51°
y
21°
18°
51 + 90 + x = 180
18 + 21 + y = 180
141 + x = 180
39 + y = 180
x
x = 39°
y = 141°
EXAMPLE 2
The triangle shown has approximate side lengths of 5 cm, 5.8 cm,
and 3 cm. Match each side with its correct length.
A
© Houghton Mifflin Harcourt Publishing Company
31°
AB = 5 cm
59°
C
BC = 3 cm
The shortest side is opposite the smallest angle.
AC = 5.8 cm
The longest side is opposite the largest angle.
B
EXERCISES
Tell whether a triangle can have sides with the given lengths. If it
cannot, give an inequality that shows why not. (Lesson 15.1)
1. 5 in., 12 in., 13 in.
2. 4.5 ft, 5.5 ft, 11 ft
Unit 5
473
Find each missing angle measure. Classify each triangle as acute,
obtuse, or right. (Lesson 15.2)
3.
4.
82°
41°
x
38°
49°
x
Match each side length with its correct measure. Classify each
triangle as scalene, isosceles, or equilateral. (Lesson 15.3)
5. The side lengths of triangle ABC are 6.4 ft, 10 ft, and 6.4 ft.
A
AB =
102°
BC =
39°
AC =
39°
B
C
6. The side length of ZX is 17 cm.
X
XY =
60°
YZ =
60°
60°
Z
?
16
16
Area and Volume
Equations
ESSENTIAL QUESTION
How can you use area and volume equations to solve real-world
problems?
4 in.
EXAMPLE 1
Find the area of the trapezoid.
A = _12 (h) (b1 + b2)
10 in.
A = _12 (10) (7 + 4)
A = 55 in2
474
Unit 5
7 in.
Key Vocabulary
parallelogram
(paralelogramo)
rhombus (rombo)
trapezoid (trapecio)
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MODULE
MODULE
Y
EXAMPLE 2
A triangular sail for a sailboat has a height of 30 feet and an area of
330 square feet. Find the base length of the sail.
A = _12 bh
()
330 = _12 30b
b = 22 ft
EXAMPLE 3
A cubic centimeter of gold weighs approximately 19.32 grams. Find
the weight of a brick of gold that has a height of 6 centimeters,
width of 3 centimeters, and length of 8 centimeters.
V = lwh
8 cm
V = 8 (3) (6)
V = 144 cm3
6 cm
3 cm
The weight of the gold is 144 × 19.32 grams,
which is 2,782.08 grams.
EXERCISES
Find the area of each figure. (Lessons 16.1, 16.2)
24 in.
1.
2.
12 in.
8 ft
8 ft
Find the missing measurement. (Lesson 16.3)
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3.
11 m
h
b
4.
4 mm
A = 62.5 m2
A = 26 mm2
14 m
Find the volume of each rectangular prism. (Lesson 16.4)
5.
6. A rectangular prism with a width of
7 units, a length of 8 units, and a height
6 in.
8 in.
20 in.
of 2 units
Unit 5
475
7. Jelani is ordering a piece of glass in the shape of a trapezoid to
create a patio table top. Each square foot of glass costs $25. The
trapezoid has base lengths of 5 feet and 3 feet and a height of
4 feet. Find the cost of the glass. (Lesson 16.1)
Unit 5 Performance Tasks
1.
Theater Set
Construction Ahmed and Karina are
building scenery of the Egyptian pyramids 1 ft
out of plywood for a community play. The
pyramids are represented by triangles on
a rectangular base. The diagram shows the
measurements of the piece of scenery.
CAREERS IN MATH
1.5 ft
0.75 ft
1.5 ft
2 ft
1 ft
0.75 ft
5 ft
a. They have one sheet of plywood, 3 ft by 6 ft. Will they be able to
make the piece using this one sheet? Explain.
b. How many square feet of plywood is in the completed piece? Show
your work.
2. Cassandra is making a design for a logo. One part
of the design is a triangle with two congruent sides.
She must draw the triangle with at least one side
with length 6 centimeters, and at least one side
with length 4 centimeters. Sketch two possible
figures that Cassandra could use. Label the side
lengths in both figures.
476
Unit 5
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c. The pyramids (the triangles) will be painted gray, and the base (the
rectangle) will be painted black. How much of each paint color will
they use, if one quart covers 45 square feet? Only one side of the
model needs to be painted, but two coats of paint will be needed.
Show your work. Round to the nearest hundredth of a square foot.
Personal
Math Trainer
UNIT 5 MIXED REVIEW
Texas Test Prep
Selected Response
1. Part of a large wooden art project will be
a triangle formed by joining three boards
together. The artist has four boards that
measure 16 feet, 11 feet, 7 feet, and 3 feet.
Which board could not be used with two of
the others to form a triangle?
my.hrw.com
Online
Assessment and
Intervention
4. What is the area of the triangle shown
below?
17 in.
26 in.
A the 3-foot board
A 110.5 square inches
B the 7-foot board
B 221 square inches
C the 11-foot board
C 442 square inches
D the 16-foot board
2. Which of these could be the value of x in
the triangle below?
A
D 884 square inches
5. The trapezoid below has an area of
475 square meters.
28 m
53°
38
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18 m
85°
42°
C
26
B
5x
A 5
C 8
B 6
D 10
3. What is the area of a trapezoid that
has bases measuring 19 centimeters
and 23 centimeters, and a height of
14 centimeters?
A 105 square centimeters
B 266 square centimeters
C 294 square centimeters
D 322 square centimeters
Which equation could you solve to find the
height of the trapezoid?
A 23h = 475
B 252h = 475
C 46h = 475
D 504h = 475
6. A rectangular prism has a volume of 1,500
cubic centimeters. It has a length of 34
centimeters and a width of 22 centimeters.
Which equation could be solved to find the
height of the rectangular prism?
A 374h = 1,500
B 28h = 1,500
C 748h = 1,500
D 56h = 1,500
Unit 5
477
7. Which expression represents the sum of 59
and x?
A 59 + x
B 59 ÷ x
Hot !
Tip
It is helpful to draw or redraw
a figure. Answers to geometry
problems may become clearer as
you redraw the figure.
11. What is the measure, in degrees, of the
missing angle in the triangle below?
C 59x
D 59 - x
Q
8. Which number has more than two factors?
39°
A 19
B 23
C 25
D 29
66°
S
9. Which of the following statements about
rational numbers is not correct?
numbers.
B All integers are also rational numbers.
C All rational numbers can be written in
the form _a where b ≠ 0.
b
D Rational numbers cannot be negative.
Gridded Response
10. What is the measure of the missing angle in
a triangle that contains angle measures of
37° and 59°?
.
478
Unit 5
.
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12. Janice wants to buy carpet for a trapezoidshaped room. The bases of the trapezoid
are 12 feet and 14 feet, and the height is
15 feet. If the carpet she likes is $5.50 per
square foot, how much will new carpet for
the room cost in dollars?
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A All whole numbers are also rational
R