10-1
10-1
Areas of Parallelograms
and Triangles
1. Plan
Objectives
1
2
To find the area of a
parallelogram
To find the area of a triangle
Examples
1
2
3
4
Finding the Area of a
Parallelogram
Finding a Missing Dimension
Finding the Area of a Triangle
Real-World Connection
What You’ll Learn
• To find the area of a
parallelogram
• To find the area of a
triangle
. . . And Why
To find the force of wind
against the side of a building,
as in Example 4
Math Background
The area of, or number of square
units covered by, a quadrilateral,
can be described algebraically as
the product of its base and height.
You can demonstrate this by transforming, cutting, and pasting
sections of the quadrilateral to
form a rectangle. Base and height
also appear in the formulas for
the area of a triangle (half of a
parallelogram) and the lateral
surface areas of prisms and
pyramids. Because of this, the
area of a rectangle is often given
in terms of base and height, not
length and width.
GO for Help
Check Skills You’ll Need
Find the area of each figure. 1. 25
1. a square with 5-cm sides
cm2
2. 28
in.2
4.
3
2
Lesson 1-9
ft2
2. a rectangle with base 4 in. and height 7 in.
3. a 4.6 m-by-2.5 m rectangle 4. a rectangle with length 3 ft and width 12 ft
11.5 m2
Each rectangle is divided into two congruent triangles. Find the area of
each triangle.
8 units2
6.
2 units2
7.
5.
6 units2
New Vocabulary • base of a parallelogram • altitude of a parallelogram
• height of a parallelogram • base of a triangle
• height of a triangle
1
Area of a Parallelogram
The diagrams at the top of page 532 show that a parallelogram with the same base
and height as a rectangle has the same area as the rectangle.
Key Concepts
Theorem 10-1
Area of a Rectangle
The area of a rectangle is the product of its
base and height.
A = bh
More Math Background: p. 530C
Theorem 10-2
b
Area of a Parallelogram
The area of a parallelogram is the product of
a base and the corresponding height.
Lesson Planning and
Resources
h
A = bh
h
b
See p. 530E for a list of the
resources that support this lesson.
Vocabulary Tip
PowerPoint
Bell Ringer Practice
The term base is used to
represent both a segment
and its length.
A base of a parallelogram is any of its sides. The corresponding altitude is a
segment perpendicular to the line containing that base, drawn from the side
opposite the base. The height is the length of an altitude.
Check Skills You’ll Need
Altitude
Base
For intervention, direct students to:
Finding the Area of a Rectangle
Lesson 1-9: Example 4
Extra Skills, Word Problems, Proof
Practice, Ch. 1
534
Chapter 10 Area
Special Needs
Below Level
L1
Students may not be familiar with labeling the sides
of a rectangle as height and base. Point out that a
rectangle is a parallelogram with four right angles,
and the height of a rectangle is always equal to
a side.
534
learning style: verbal
L2
After students find the area in Example 2
algebraically, have them count the number of squares
and parts of squares and compare their answers with
15 square units.
learning style: visual
1
nline
2. Teach
Finding the Area of a Parallelogram
EXAMPLE
Find the area of each parallelogram.
a.
b.
4.5 in.
4.6 cm
4 in.
Guided Instruction
3.5 cm
1
2 cm
5 in.
You are given each height. Choose the corresponding side to use as the base.
A = bh
Visit: PHSchool.com
Web Code: aue-0775
The area is 20
Quick Check
= 2(3.5) = 7
Substitute.
in.2.
2
The area is 7 cm2.
Finding a Missing Dimension
EXAMPLE
For $ABCD, find CF to the nearest tenth.
F
First, find the area of $ABCD. Then use the
area formula a second time to find CF.
13 in.
= 10(12) = 120
C
The area of $ABCD is 120 in.2.
For: Area Activity
Use: Interactive Textbook, 9-1
A
A = bh
120 = 13(CF)
Use base AD and height CF.
E
10 in.
B
8m
CF = 120
13 < 9.2
CF is about 9.2 in.
Quick Check
2
1
Additional Examples
1 Find the area of the
parallelogram.
12 in.
Use base AB and height DE.
Error Prevention
PowerPoint
D
A = bh
EXAMPLE
If students think the base that
corresponds to height CF is AF
rather than AD, remind them that
the base is always a side of the
parallelogram.
1 Find the area of a parallelogram with base 12 m and height 9 m. 108 m2
2
Teaching Tip
Make sure that students understand that 5 in. is the measure
of the entire base in part a.
A = bh
= 5(4) = 20
EXAMPLE
10.5 m
12 m
96 m2
2 A parallelogram has sides 15 cm and 18 cm. The height corresponding to a 15-cm
base is 9 cm. Find the height corresponding to an 18-cm base. 7.5 cm
Area of a Triangle
2 A parallelogram has 9-in.
and 18-in. sides. The height
corresponding to the 9-in.
base is 15 in. Find the height
corresponding to the 18-in. base.
7.5 in.
You can rotate a triangle about the midpoint of a side to form a parallelogram.
M
h
h
b
b
The area of the triangle is half the area of the parallelogram.
Key Concepts
Theorem 10-3
Area of a Triangle
The area of a triangle is half the product
of a base and the corresponding height.
A = 12 bh
h
b
Lesson 10-1 Areas of Parallelograms and Triangles
Advanced Learners
535
English Language Learners ELL
L4
After Examples 1 and 2, have students explore
possible areas for a parallelogram with side lengths
12 cm and 10 cm. They should justify their conclusions.
learning style: verbal
Have students repeat aloud the formulas for the area
of a rectangle, parallelogram, and triangle. Note that
in “area equals one-half base times height,” the base
and height must be perpendicular segments.
learning style: verbal
535
Guided Instruction
A base of a triangle is any of its sides. The corresponding height is the length of
the altitude to the line containing that base.
Tactile Learners
Have students cut out two copies
of a triangle and then join the
triangles in three ways to form
parallelograms. Students can
readily see that the area of each
parallelogram is twice the area
of one triangle.
4
EXAMPLE
3
Find the area of the triangle.
A = 12 bh
= 12(10)(6.4) = 32
Connection
to Algebra
Quick Check
4 ft
13 cm
5 cm
Real-World
EXAMPLE
Connection
Structural Design When designing a building, you must be sure that the building
can withstand hurricane-force winds, which have a velocity of 73 mi/h or more. The
formula F = 0.004Av2 gives the force F in pounds exerted by a wind blowing
against a flat surface. A is the area of the surface in square feet, and v is the wind
velocity in miles per hour.
3 Find the area of XYZ.
X
How much force is exerted by a 73 mi/h wind blowing
directly against the side of the building shown here?
31 cm
6 ft
Find the area of the side of the building.
13 cm
Z
Real-World
cm2
4 The front of a garage is a
square 15 ft on each side with a
triangular roof above the square.
The height of the triangular roof is
10.6 ft. To the nearest hundred,
how much force is exerted by an
80 mi/h wind blowing directly
against the front of the garage?
Use the formula F = 0.004Av2.
about 7800 lb
Resources
• Daily Notetaking Guide 10-1
L3
• Daily Notetaking Guide 10-1—
L1
Adapted Instruction
12 ft
triangle area = 21 bh = 12(20)6 = 60 ft2
rectangle area = bh = 20(12) = 240 ft2
20 ft
area of the side = 60 + 240 = 300 ft2
Use the area of the side of the building and the
velocity of the wind to find the force.
F = 0.004Av2
= 0.004(300)(73)2
Use the formula for force.
Substitute 300 for A and 73 for v.
= 6394.8
The force is about 6400 lb, or 3.2 tons.
Quick Check
4 Critical Thinking Suppose the bases of the rectangle and triangle in the building
above are doubled to 40 ft, but the height of each figure remains the same. How is
the force of the wind against the side of the building affected?
The force is doubled.
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
Practice by Example
Closure
An isosceles trapezoid has 10-m
and 20-m bases, and the height is
8 m. Find the area. Hint: Draw a
diagonal. 120 m2
Connection
In 1992 this building in
Homestead, Florida,
succumbed to the 145 mi/h
winds of Hurricane Andrew.
A
GO for
Help
Example 1
(page 535)
Find the area of each parallelogram.
1.
15 cm
20.3 m2
3.
12 cm
20 cm
Chapter 10 Area
3.5 m
26.79 in.2
2.
240 cm2
536
536
10 ft
3 Find the area of the triangle.
30 cm2
4
Additional Examples
195
Substitute and simplify.
12 cm
PowerPoint
30 cm
6.4 ft
The area of the triangle is 32 ft2.
Have students carefully examine
the formula. Ask: What happens to
force F if surface area A doubles?
Force doubles. What happens to
force F if wind velocity v doubles?
Force quadruples.
Y
Finding the Area of a Triangle
EXAMPLE
5.7 in.
6 in.
5.8 m
4.7
4m
Example 2
(page 535)
4. 11.2
5.
h
0.3
14
8
Example 3
(page 536)
3. Practice
Find the value of h for each parallelogram.
10
0.24
0.5
6.
13
h
12
h
18
0.4
1 A B 1-6, 11-16, 24-28
8
1613
2 A B
Find the area of each triangle.
7.
8.
14 m2
5.7 m
5m
9.
4.5 yd
6 yd
C Challenge
Example 4
(page 536)
10b. Find the entire area
of the lot and
subtract the area
for the flowers.
10c. 1550 – 160 ≠
1390 ft2
B
Apply Your Skills
3m
7-10, 17-23, 29-36
37-39
3 ft2
3 ft
4m
4m
Assignment Guide
Test Prep
Mixed Review
40-44
45-55
7.5 yd
13.5 yd2
10. Landscaping Taisha’s Bakery has a plan for
a 50 ft-by-31 ft parking lot. The four parking
spaces are congruent parallelograms, the driving
region is a rectangle, and the two unpaved areas
for flowers are congruent triangles.
a. Find the area of the surface to be paved by
adding the areas of the driving region and
the four parking spaces. 1390 ft2
b. Describe another method for finding the
area of the surface to be paved.
c. Use your method from part (b) to find
the area. Then compare answers from
parts (a) and (b) to check your work.
2 ft
Homework Quick Check
2 ft
To check students’ understanding
of key skills and concepts, go over
Exercises 2, 10, 24, 29, 30.
10 ft
Exercises 1–3 Make sure that
students also square the unit
of measurement.
50 ft
Exercises 7–9 To help students
find corresponding heights and
bases, suggest that they rotate
their textbooks so that each
parallelogram has a horizontal
base and vertical height.
15 ft
31 ft
11. The area of a parallelogram is 24 in.2 and the height is 6 in. Find the
corresponding base. 4 in.
12. Multiple Choice What is the area of the figure
at the right? B
64 cm2
88 cm2
2
96 cm
112 cm2
13. An isosceles right triangle has area of 98 cm2.
Find the length of each leg. 14 cm
Exercise 12 Ask: How do you
know that part of the figure is a
square? four right angles and all
sides are congruent
14 cm
8 cm
8 cm
x 2 14. Algebra In a triangle, a base and a corresponding height are in the ratio 3 : 2.
The area is 108 in.2. Find the base and the corresponding height. 18 in.; 12 in.
GO
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-1001
15. Technology Ki used
geometry software to
create the figure at the
k
C
D
*right.
) She constructed
AB and a point C not on
* )
AB .Then she constructed
* )
line k parallel to AB
through point C. Next,
A
B
Ki constructed point D
on line k as well as AD
and BD. She dragged point D along line k to manipulate #ABD. How does the
area of #ABD change? Explain. See left.
16. Open-Ended Using graph paper, draw an acute triangle, an obtuse triangle, and
a right triangle, each with area 12 units2. See margin.
Lesson 10-1 Areas of Parallelograms and Triangles
537
GPS Guided Problem Solving
L3
L4
Enrichment
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
L3
Date
Practice 10-1
Space Figures and Nets
1. Choose the nets that will fold to make a cube.
A.
B.
C.
D.
Draw a net for each figure. Label each net with its appropriate dimensions.
2.
16 cm
7 cm
2 cm
3.
8 cm
4.
1 cm
2 cm
32 cm
1 cm
40 cm
Match each three-dimensional figure with its net.
5.
6.
A.
7.
B.
8.
C.
D.
9. Choose the nets that will fold to make a pyramid with a square base.
A.
B.
C.
D.
© Pearson Education, Inc. All rights reserved.
15. The area does not
change; the height and
base AB do not change.
Use Euler’s Formula to find the missing number.
10. Faces: 5
Edges: 7
Vertices: 5
11. Faces: 7
Edges: 9
Vertices: 6
12. Faces: 8
Edges: 18
Vertices: 7
537
4. Assess & Reteach
17. 15 units2
18. 6
Find the area of each figure.
units2
19. 6 units2
PowerPoint
Lesson Quiz
20. 12
units2
21. 27 units2
1. Find the area of the
parallelogram.
22. 3
18. #BDJ
4
19. #DKJ
20. $BDKJ
2
21. $ADKF
22. #BCJ
2
4
B
6
K
C
8
D
10
12
x
In Exercises 24–27, (a) graph the lines and (b) find the area of the triangle enclosed
by the lines. 24–27. See margin.
25. y = x + 2, y = 2, x = 6
26. y = -12 x + 3, y = 0, x = -2
27. y = 34 x - 2, y = -2, x = 4
28. Find the area of the yellow triangular patch in the large field in the photo at the
left. It has a base of 60 yd and a height of 140 yd. 4200 yd2
12 ft
150 ft2
29. Probability Ann drew these three figures on a grid. A fly lands at random at a
point on the grid.
2. Find the area of XYZW with
base 4 units and height 6 units.
24 square units
3. A parallelogram has 6-cm
and 8-cm sides. The height
corresponding to the 8-cm
base is 4.5 cm. Find the height
corresponding to the 6-cm
base. 6 cm
a. Writing Is the fly more likely to land on one of the figures or on the blank
grid? Explain. Blank grid; area is 84 units2 while figures are 36 units2.
b. Suppose you know the fly lands on one of the figures. Is the fly more likely to
land on one figure than on another? Explain.
No; the figures have the same area.
Coordinate Geometry Find the area of a polygon with the given vertices.
60 units2
28 units2
30. A(3, 9), B(8, 9), C(2, -3), D(-3, -3) 31. E(1, 1), F(4, 5), G(11, 5), H(8, 1)
4. Find the area of RST.
R
Exercise 28
11 m
T
15 m2
32. D(0, 0), E(2, 4), F(6, 4), G(6, 0)
20 units2
Find the area of each figure.
5. A rectangular flag is divided
into four regions by its
diagonals. Two of the regions
are shaded. Find the total area
of the shaded regions.
34.
35.
25 ft
33. K(-7, -2), L(-7, 6), M(1, 6), N(7, -2)
88 units2
36.
15 cm
200 m
21 cm
120 m
25 ft
22 in.
17 in.
C
187
A
O
GPS 24. y = x, x = 0, y = 7
6m
J
F
units2
15 ft
S
17. $ABJF
23. ADJF 21 units2
10 ft
5m
y
Challenge
in.2
40 m
60 m
25 ft
20 cm
12,800 m2
312.5 ft2
525 cm2
History The ancient Greek mathematician Heron is most famous for this formula
for the area of a triangle in terms of the lengths of its sides a, b, and c.
A = "s(s 2 a)(s 2 b)(s 2 c) , where s = 12(a + b + c)
Alternative Assessment
Use Heron’s Formula and a calculator to find the area of each triangle. Round your
answer to the nearest whole number.
Have each student draw and label
a triangle and a parallelogram,
each with an area of 40 in.2 Have
them write a paragraph explaining
how they calculated the area of
each figure.
37. a = 8 in., b = 9 in., c = 10 in. 34 in.2 38. a = 15 m, b = 17 m, c = 21 m 126 m2
39. a. Use Heron’s Formula to find the area of this triangle.
b. Verify your answer to part (a) by using the
54 in.2
formula A = 12 bh. 54 in.2
538
4
6
9 in.
12 in.
Chapter 10 Area
16. Answers may vary.
Sample:
538
15 in.
24. a
6
x0
4
4
b. 25 units2
y7
y
yx
4
2
6
6
x
O
2
4
6
Test Prep
Test Prep
Multiple Choice
Resources
40. The lengths of the sides of a right triangle are 10 in., 24 in., and 26 in.
What is the area of the triangle? B
A. 116 in.2
B. 120 in.2
C. 130 in.2
D. 156 in.2
41. What is the area of $ABCD at the right? G
F. 32 in.2
G. 64 in.2
2
H. 91.2 in.
J. 45.6 in.2
D
C
8 in.
42. A parallelogram has adjacent sides of 176 ft
A
and 312 ft. The altitude to the shorter side is
290 ft. What is the area of the parallelogram? A
A. 51,040 ft2
B. 51,352 ft2
C. 54,912 ft2
11.3 in.
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 593
• Test-Taking Strategies, p. 588
• Test-Taking Strategies with
Transparencies
B
8 in.
D. 55,202 ft2
43. The perimeter of an equilateral triangle is 60 m. Its height is 17.3 m. What
is its area? F
F. 173 m2
G. 200 m2
H. 348 m2
J. 1044 m2
Short Response
44. a. For $ABCD, explain how to
determine the length of an
altitude drawn to base AB.
b. Find the area of $ABCD.
a-b. See margin.
y
2
D (-3, 2)
C (2, 2)
O
x
2
A (-1, -3)
B (4, -3)
44. [2] a. It is the distance
between y ≠ 2 and
y ≠ –3, so
2 – (–3) ≠ 5.
Mixed Review
b. 25 units2
Lesson 9-7
GO for
Help
List the symmetries in each tessellation. 45–46. See back of book.
45.
[1] incorrect explanation
OR incorrect answer
46.
53.
D
A
Lesson 4-5
Lesson 3-8
The base of the isosceles triangle is a side
of a regular pentagon PENTA. Find the
measure of each angle.
47. &APE 108
48. &APN 72
49. &PAN 72
50. &PNA 36
51. &EPN 36
52. &ANT 36
B
N
E
T
54.
P
A
G
53–55. See margin.
Use a compass and straightedge for the following constructions.
*
)
*
)
E
53. Draw a segment and label it AB. Construct AD so that AD ' AB at point A.
*
)
*
)
54. Draw a segment. Label it EF. Construct a line GH so that GH 6 EF.
* )
55. Draw a segment and label it KL. Draw a point X not on
KL
*
)) .
(
H
55.
F
X
Construct a perpendicular from point X to KL or to KL .
Lesson 10-1 Areas of Parallelograms and Triangles
lesson quiz, PHSchool.com, Web Code: aua-1001
25. a
26. a
y
4
x
x 2
y
y2
4
x6
2
6
27. a
y
y 12 x 3
2
O
y0 4
539
K
L
y
y 34 x 2
1
O
1
3
x
x4
y 2
x
x
O
2
4
539