 # 11-2

```11-2
MAIN IDEA
Find the areas of
triangles and trapezoids.
Area of Triangles
and Trapezoids
1. 24 sq units
2. They are congruent.
• Draw a parallelogram with a base of
6 units and a height of 4 units.
• Draw a diagonal as shown.
Math Online
• Cut out the parallelogram.
glencoe.com
1. What is the area of the parallelogram?
• Concepts In Motion
• Extra Examples
• Personal Tutor
• Self-Check Quiz
2. Cut along the diagonal. What is true about the triangles formed?
3. What is the area of each triangle?
12 sq units
4. If the area of a parallelogram is bh, then write an expression for the
area A of each of the two congruent triangles that form the
parallelogram. A = 1 bh
_
2
You can find the area of a triangle by using the base and height.
The base of a
triangle can be
any of its sides.
h
The height is the perpendicular
distance from the vertex
opposite the base to the line
containing the base.
b
Area of a Triangle
Words
Symbols
The area A of a triangle
equals half the product of
its base b and height h.
Key Concept
Model
_1
A = bh
h
b
2
Find the Area of a Triangle
1 Find the area of the triangle shown.
_
1
Estimate A = (10)(7) or 35
2
1
A=_
bh
2
1 ( )( )
A=_
10 6.5
2
A = 32.5
Area of a triangle
Replace b with 10 and h with 6.5.
Multiply.
6.5 m
10 m
The area of the triangle is 32.5 square meters.
Check for Reasonableness 32.5 ≈ 35 ✔
578
Chapter 11 Measurement: Two- and Three-Dimensional Figures
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Find the area of each triangle. Round to the nearest tenth
if necessary.
a.
b.
77 ft 2
30 in 2
8 in.
11 ft
1
7_
in.
14 ft
2
A trapezoid has two bases, b 1 and b 2. The bases are always the two sides
that are parallel. The height of a trapezoid is the perpendicular distance
between the bases.
Area of a Trapezoid
b sub 1. Read b 2 as b
sub 2. The subscripts
mean that b 1 and b 2
represent different
variables.
Words
Key Concept
The area A of a trapezoid
equals half the product of
the height h and the sum
of the bases b 1 + b 2.
b1
Model
h
_1
b2
A = h(b 1 + b 2)
Symbols
2
Find the Area of a Trapezoid
2 Find the area of the trapezoid.
5 in.
The bases are 5 inches and 12 inches.
The height is 7 inches.
1
A=_
h(b1 + b2 )
2
_
A = 1 (7 ) (5 + 12 )
2
1 ( )( )
A=_
7 17
2
A = 59.5
7 in.
Area of a trapezoid
12 in.
Replace h with 7, b 1 with 5, and b 2 with 12.
Multiply.
The area of the trapezoid is 59.5 square inches.
Find the area of each trapezoid. Round to the nearest tenth
if necessary.
c.
2.5 m
14.6 m 2
d.
1 ft
0.2 ft 2
0.3 ft
4m
4.8 m
0.5 ft
Lesson 11-2 Area of Triangles and Trapezoids
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3 GEOGRAPHY The shape of Iron County, Michigan, resembles a
trapezoid. Find the approximate area of this county.
Keweenaw
Houghton
Ontonagon
Gogebic
22 mi
Baraga
42 mi
Iron
Marquette
Alger
34 mi
Dickinson
Luce
Schoolcraft
Chippewa
Mackinac
Delta
Menominee
1
A=_
h(b 1 + b 2)
2
_
A = 1 (42)(22 + 34)
2
_
A = 1 (42)(56)
2
A = 1,176
Area of a trapezoid
Replace h with 42, b 1 with 22, and b 2 with 34.
Multiply.
The area of Iron County is approximately 1,176 square miles.
280 mi
e. GEOGRAPHY The shape of the state
ARKANSAS
of Arkansas resembles a trapezoid.
Find the approximate area of Arkansas.
235 mi
Little Rock
210 mi
Examples 1, 2
(pp. 578–579)
Find the area of each figure. Round to the nearest tenth if necessary.
1.
2.
3.
16.5 m
3 in.
4 in.
Example 3
(p. 580)
580
8 ft
15.6 ft
12.8 m
4. HOCKEY In the National Hockey
League, goaltenders can play
the puck behind the goal line
only in a trapezoid-shaped area,
as shown at the right. Find the
area of the trapezoid.
7 ft
18 ft
11 ft
28 ft
Chapter 11 Measurement: Two- and Three-Dimensional Figures
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HOMEWORK
For
Exercises
5, 6, 9
7, 8, 10
11, 12
HELP
See
Examples
1
2
3
Find the area of each figure. Round to the nearest tenth if necessary.
5.
147 in 2
6.
38.4 mm 2
8 mm
14 in.
21 in.
7.
1.1 cm
2 cm
9.6 mm
3.4 cm
4.5 cm 2
Exercise Levels
A: 5–12
B: 13–20
C: 21–23
17.75 m
8.
9.
183.7 in 2
112 m 2
10 ft
1
22 in.
8m
15 ft
10.
8 2 ft
10.25 m
16.7 in.
23 ft
161.5 ft 2
11. GEOGRAPHY The shape of Idaho is
roughly triangular with a base of
380 miles and a height of 500 miles.
Find the approximate area of Idaho.
12. ALGEBRA Find the area of a trapezoid
with bases 13 inches and 15 inches and
a height of 7 inches. 98 in 2
500 mi
IDAHO
380 mi
ALGEBRA Find the height of each figure.
13.
174 ft
x ft
264 yd
14.
125 ft
x yd
2
A = 11,500 ft
184 ft
130 yd
2
A = 29,185 yd 145 yd
185 yd
Draw and label each figure. Then find the area. 15–17. See margin.
15. a triangle with no right angles and an area less than 12 square centimeters
16. a trapezoid with a right angle and an area greater than 40 square meters
17. a trapezoid with no right angles and an area less than 25 square feet
23 in.
★ 18. TENTS A play tent is shown at the right.
How much fabric was used to make
the front and back of the play tent? 1,904 in 2
32 in.
36.5 in.
LANDSCAPING For Exercises 19 and 20, use
the diagram that shows the lawn that
surrounds an office building.
★ 19. What is the area of the lawn? 6,500 ft 2
EXTRA
PRACTICE
See pages 697, 714.
20. If one bag of grass seed covers
2,000 square feet, how many bags
are needed to seed the lawn? 4 bags
100 ft
50 ft
80 ft
62 ft
140 ft
Lesson 11-2 Area of Triangles and Trapezoids
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H.O.T. Problems
21. Sample
of the areas is
the square of the
ratio of the
bases.
21. CHALLENGE Triangle ABC has a base of 4 units and a height of 8 units.
Triangle DEF has twice the base and height. Describe how the ratio
of the bases is related to the ratio of the areas.
h
to explain how to estimate the height h of the
trapezoid shown if the area is 235.5 in 2. See margin.
23.
25. SHORT RESPONSE Randy was hired to
J
F
12 ft
x ft
18 ft G
26.75 in.
WR ITING IN MATH Describe the relationship between the area of a
parallelogram and the area of a triangle with the same height and base.
The area of a triangle is half the area of a parallelogram with the same base
and height, because two of these triangles make up the parallelogram.
24. FGH and JKM are similar.
H
19.95 in.
22. REASONING Apply what you know about rounding
M
24 ft
put in sod on a piece of land shaped
like a trapezoid as shown. How many
square feet of sod are needed?
11,000 ft2
K
Which choice shows the equations that
can be used to find the area of FGH?
C
18
12
1(
A _
=_
and then _
18x)
50 ft
120 ft
80 ft
2
24
x
18
x
B _
=_
and then 18x
24
12
18
x
1(
C _
=_
and then _
18x)
2
24
12
18
12
D _
=_
and then 18x
24
x
140 ft
225 ft
26. MEASUREMENT Find the area of a parallelogram having a base of 2.3 inches
and a height of 1.6 inches. Round to the nearest tenth.
(Lesson 11-1)
3.7 in 2
★27. GEOMETRY Graph JLK with vertices J(-1, -4), L(3, -2) and K(1, 1), and
its reflection over the x-axis. Write the ordered pairs for the vertices of
the new figure. (Lesson 10-10) See margin.
Find each number. Round to the nearest tenth if necessary.
28. What number is 56% of 600? 336
30. 72 is 45% of what number? 160
(Lesson 7-2)
29. 24.5 is what percent of 98? 25%
31. 62.5% of 250 is what number? 156.3
PREREQUISITE SKILL Use a calculator to find each product to the nearest tenth.
(Lesson 1-4)
32. π · 13
582
40.8
33. π · 29
91.1
34. π · 16 2
804.2
35. π · 4.8 2
72.4
Chapter 11 Measurement: Two- and Three-Dimensional Figures
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``` # 3.3 Area and Perimeter of Parallelograms on the Coordinate Plane 