Download Chapter 13

Topic 13
Area
TOPIC OVERVIEW
VOCABULARY
13-1Areas of Parallelograms and Triangles
13-2Areas of Trapezoids, Rhombuses,
and Kites
13-3Areas of Regular Polygons
13-4Perimeters and Areas of Similar
Figures
13-5Trigonometry and Area
DIGITAL
APPS
English/Spanish Vocabulary Audio Online:
EnglishSpanish
altitude of a parallelogram, p. 520
altura de un paralelogramo
apothem, p. 532apotema
base of a parallelogram, p. 520
base de un paralelogramo
base of a triangle, p. 520
base du un triángulo
center of a regular polygon, p. 532
centro de un polígono regular
composite figure, p. 520 figura compuesta
height of a parallelogram, p. 520
altura de un paralelogramo
height of a trapezoid, p. 526
altura de un trapecio
height of a triangle, p. 520
altura de un triángulo
radius of a regular polygon, p. 532
radio de un polígono regular
PRINT and eBook
Access Your Homework . . .
Online homework
You can do all of your homework online with built-in examples and
“Show Me How” support! When you log in to your account, you’ll see
the homework your teacher has assigned you.
Your Digital
Resources
PearsonTEXAS.com
Homework Tutor app
Do your homework anywhere! You can access the Practice and
Application Exercises, as well as Virtual Nerd tutorials, with this
Homework Tutor app, available on any mobile device.
STUDENT TEXT AND Homework Helper
Access the Practice and Application Exercises that you are assigned for
homework in the Student Text and Homework Helper, which is also
available as an electronic book.
516
Topic 13 Area
3--Act Math
The Great
Enlargement
Today, many people take
photos with a digital camera
or a smartphone. Digital
photographs are easy to edit,
store, and share with others.
Now, printing digital photos
is becoming easier. More
and more new printers have
e-printing capabilities, so
people can print directly from
their cameras or smartphones.
This 3-Act Math task will
make you think twice about
printing photos!
Scan page to see a video
for this 3-Act Math Task.
If You Need Help . . .
Vocabulary Online
You’ll find definitions of math
terms in both English and
Spanish. All of the terms have
audio support.
Learning Animations
You can also access all of the
stepped-out learning animations
that you studied in class.
Interactive Math tools
These interactive math tools
give you opportunities to
explore in greater depth
key concepts to help build
understanding.
Interactive exploration
You’ll have access to a robust
assortment of interactive
explorations, including
interactive concept explorations,
dynamic activitites, and topiclevel exploration activities.
Student Companion
Refer to your notes and
solutions in your Student
Companion. Remember that
your Student Companion is also
available as an ACTIVebook
accessible on any digital device.
Virtual Nerd
Not sure how to do some of
the practice exercises? Check
out the Virtual Nerd videos
for stepped-out, multi-level
instructional support.
PearsonTEXAS.com
517
Activity Lab
Use With Lessons 13-1 and 13-2
Transforming to Find Area
teks (3)(A), (1)(D)
You can use transformations to find formulas for the areas of polygons. In these
activities, you will cut polygons into pieces and use the pieces to form different
polygons.
1
Step 1 Count and record the number of units in the base and the
height of the parallelogram at the right.
Step 2 Copy the parallelogram onto grid paper.
Step 3 Cut out the parallelogram. Then cut it into two pieces as shown.
Step 4 Translate the triangle to the right through a distance equal to the
base of the parallelogram.
The translation results in a rectangle. Since their pieces are congruent, the
parallelogram and rectangle have the same area.
1.How many units are in the base of the rectangle? The height of
the rectangle?
hsm11gmse_1001a_t09484
2.How do the base and height of the rectangle compare to the base and
height of the parallelogram?
3.Write the formula for the area of the rectangle. Explain how you can use this
formula to find the area of a parallelogram.
2
hsm11gmse_1001a_t09486
Step 1 Count and record the number of units in the base and the height
of the triangle at the right.
Step 2 Copy the triangle onto grid paper. Mark the midpoints A and B and
draw midsegment AB.
A
B
Step 3 Cut out the triangle. Then cut it along AB.
Step 4 Rotate the small triangle 180° about the point B.
The bottom part of the triangle and the image of the top part form a
parallelogram.
4.How many units are in the base of the parallelogram? The height
of the parallelogram?
B
A
hsm11gmse_1001a_t09490
continued on next page ▶
518
Activity Lab Transforming to Find Area
hsm11gmse_1001a_t09491
Activity Lab
continued
5.How do the base and height of the parallelogram compare to the base and height
of the original triangle? Write an expression for the height of the parallelogram in
terms of the height h of the triangle.
6.Write your formula for the area of a parallelogram from Activity 1. Substitute the
expression you wrote for the height of the parallelogram into this formula. You
now have a formula for the area of a triangle.
3
Step 1 Count and record the bases and height of the trapezoid at the right.
Step 2 Copy the trapezoid. Mark the midpoints M and N, and draw
midsegment MN .
Step 3 Cut out the trapezoid. Then cut it along MN .
M
N
Step 4 Transform the trapezoid into a parallelogram.
7. What transformation did you apply to form a parallelogram?
8. What is an expression for the base of the parallelogram in terms of the
two bases, b1 and b2 , of the trapezoid?
hsm11gmse_1001a_t09492
9. If h represents the height of the trapezoid, what is an expression in
terms of h for the height of the parallelogram?
10. Substitute your expressions from Questions 8 and 9 into your area formula
for a parallelogram. What is the formula for the area of a trapezoid?
Exercises
11. In Activity 2, can a different rotation of the small triangle form a
parallelogram? If so, does using that rotation change your results? Explain.
12. Make another copy of the Activity 2 triangle. Find a rotation of the entire
triangle so that the preimage and image together form a parallelogram. How
can you use the parallelogram and your formula for the area of a
parallelogram to find the formula for the area of a triangle?
13. a. In the trapezoid at the right, a cut is shown from the midpoint
of one leg to a vertex. What transformation can you apply to
the top piece to form a triangle from the trapezoid?
N
b.
Use your formula for the area of a triangle to find a formula for
the area of a trapezoid.
14. Count and record the lengths of the diagonals, d1 and d2 , of the
kite at the right. Copy and cut out the kite. Reflect half of the kite
across the line of symmetry d1 by folding the kite along d1 . Use
your formula for the area of a triangle to find a formula for the
area of a kite.
d1
d2
hsm11gmse_1001a_t09495
PearsonTEXAS.com
hsm11gmse_1001a_t09496
519
13-1 Areas of Parallelograms
and Triangles
TEKS FOCUS
VOCABULARY
TEKS (11)(B) Determine the area of
composite two-dimensional figures
comprised of a combination of triangles,
parallelograms, trapezoids, kites, regular
polygons, or sectors of circles to solve
problems using appropriate units of
measure.
•Altitude of a parallelogram – An altitude of a parallelogram is any
TEKS (1)(C) Select tools, including real
objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques,
including mental math, estimation, and
number sense as appropriate, to solve
problems.
•Composite figure – A composite figure is a combination of two or
Additional TEKS (1)(F), (11)(A)
segment perpendicular to the line containing the base, drawn from
the side opposite the base.
•Base of a parallelogram – A base of a parallelogram is any one of the
parallelogram’s sides.
•Base of a triangle – A base of a triangle is any one of the triangle’s
sides.
more figures.
•Height of a parallelogram – The height of a parallelogram is the
length of an altitude of the parallelogram.
•Height of a triangle – The height of a triangle is the length of the
altitude to the line containing that base.
•Number sense – the understanding of what numbers mean and how
they are related
ESSENTIAL UNDERSTANDING
You can find the area of a parallelogram or a triangle when you know the lengths of
its base and its height.
Key Concept Parts of a Parallelogram
Term Description
A base of a parallelogram can be any one 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.
Diagram
Altitude
Base
Key Concept Area of a Rectangle
hsm11gmse_1001_t09124
The area of a rectangle is the product of its base and height.
A = bh
h
b
520
Lesson 13-1 Areas of Parallelograms and Triangles
hsm11gmse_1001_t09119
Key Concept Area of a Parallelogram
The area of a parallelogram is the product of a base and the
corresponding height.
h
A = bh
b
Key Concept Area of a Triangle
hsm11gmse_1001_t09121
The area of a triangle is half the product of a base and the
corresponding height.
h
A = 12bh
b
Postulate 13-1 Area Addition Postulate
hsm11gmse_1001_t09140
The area of a region is equal to the sum of the areas of its nonoverlapping parts.
Problem 1
Finding the Area of a Parallelogram
What is the area of each parallelogram?
Why aren’t the sides
of the parallelogram
considered altitudes?
Altitudes must be
perpendicular to the
bases. Unless the
parallelogram is also a
rectangle, the sides are
not perpendicular to
the bases.
A B
4.6 cm
3.5 cm
4.5 in.
4 in.
2 cm
5 in.
You are given each height. Choose the corresponding side to use as the base.
A = bh
A = bh
hsm11gmse_1001_t09134
hsm11gmse_1001_t09132
= 5(4)
Substitute for b and h.
= 20
The area is
= 2(3.5)
=7
20 in.2.
The area is 7 cm2.
PearsonTEXAS.com
521
Problem 2
TEKS Process Standard (1)(F)
Finding a Missing Dimension
For ▱ABCD, what is DE to the nearest tenth?
F
9 in.
D
C
13 in.
A
What does CF
represent?
CF is an altitude of the
parallelogram when
AD and BC are used as
bases.
E
9.4 in.
B
First, find the area of ▱ABCD. Then use the area formula a second time to find DE.
A = bh
= 13(9)hsm11gmse_1001_t09136
= 117
Use base AD and height CF.
The area of ▱ABCD is 117 in.2.
A = bh
117 = 9.4(DE)
Use base AB and height DE.
117
DE = 9.4 ≈ 12.4
DE is about 12.4 in.
Problem 3
Finding the Area of a Triangle
Why do you need
to convert the base
and the height into
inches?
You must convert
them both because
you can only multiply
measurements with like
units.
Sailing You want to make a triangular sail like the one at
the right. How many square feet of material do you need?
Step 1 Convert the dimensions of the sail to inches.
+ 2 in. = 146 in.
(12 ft # 121 in.
ft )
+ 4 in. = 160 in.
(13 ft # 121 in.
ft )
Use a conversion factor.
Step 2 Find the area of the triangle.
A = 12bh
= 12 (160)(146)
Substitute 160 for b and 146 for h.
= 11,680
Simplify.
Step 3Convert 11,680 in.2 to square feet.
11,680 in.2
ft # 1 ft
1 2
# 121 in.
12 in. = 819 ft
You need 8119 ft2 of material.
522
12 ft 2 in.
Lesson 13-1 Areas of Parallelograms and Triangles
13 ft 4 in.
Problem 4
TEKS Process Standard (1)(C)
Finding the Area of a Composite Figure
Select a technique that will help you find the area of the composite
figure below. Then find the area of the figure.
3 cm
Could you divide the
composite figure
differently and still
use the Area Addition
Postulate?
Yes, but you might not be
able to use mental math
to do the calculations.
6 cm
3 cm
4 cm
You can use mental math for this problem, since the calculations are
easy enough to do in your head.
The area of each triangle is 12(3)(4) = 6.
The area of the parallelogram is (6)(4) = 24.
To find the area of the entire figure, add the areas of the two triangles and
the parallelogram.
6 + 6 + 24 = 36
NLINE
HO
ME
RK
O
The area of the composite figure is 36 cm2.
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Find the value of h for each parallelogram.
1.
2.
h
14
For additional support when
completing your homework,
go to PearsonTEXAS.com.
8
0.5
0.3
3.
13
h
0.4
10
h
12
18
Find the area of each figure.
y
hsm11gmse_1001_t09165
hsm11gmse_1001_t09162
J
K
F
hsm11gmse_1001_t09156
4
2
A
O
B
2
4
6
C
8
x
D
10
12
4.
▱ ABJF 5.△BDJ 6.△DKJ
7.
▱ BDKJ 8.▱ ADKF 9.△BCJ
10.trapezoid ADJF
hsm11gmse_1001_t09177
PearsonTEXAS.com
523
11.Apply Mathematics (1)(A) A bakery has a
50 ft-by-31 ft parking lot. The four parking
spaces are congruent parallelograms, the
driving region is a rectangle, and the two
areas for flowers are congruent triangles.
a.Find the area of the paved surface by
adding the areas of the driving region
and the four parking spaces.
31 ft
10 ft
15 ft
b.Describe another method for finding
the area of the paved surface.
50 ft
c.Use your method from part (b) to find
the area. Then compare answers from parts (a) and (b) to check your work.
12.What is the area of the figure at the right?
14 cm
A.64 cm2 C.96 cm2
B.88 cm2 D.112 cm2
8 cm
13.The area of a parallelogram is 24 in.2 , and the height is
6 in. Find the length of the corresponding base.
HSM11GMSE_1001_a09795
1st pass 12-19-08
Durke
8 cm
14.A right isosceles triangle has area 98 cm2 . Find the length of each leg.
15.Analyze Mathematical Relationships (1)(F) The area of a triangle is 108 in.2 .
hsm11gmse_1001_t09175
A base and corresponding height are in the ratio 3 : 2. Find the
length of the base
and the corresponding height.
Find the area of each figure.
16.
17.
25 ft
21 cm
18.
15 cm
25 ft
200 m
120 m
40 m
25 ft
60 m
20 cm
Select Techniques to Solve Problems (1)(C) Select a technique (such as mental
math, estimation, or number sense) to find the area of the composite figure. Then
find the area.
19.
hsm11gmse_1001_t09179
hsm11gmse_1001_t09183.ai
hsm11gmse_1001_t09181.ai
4 yd
20.
6m
10 yd
6 yd
7m
5m
20 yd
For Exercises 21 and 22, (a) graph the lines and (b) find the area of the triangle
enclosed by the lines.
21.y = - 12 x + 3, y = 0, x = -2
524
Lesson 13-1 Areas of Parallelograms and Triangles
3
22. y = 4 x - 2, y = -2, x = 4
Find the area of a polygon with the given vertices.
23.E(1, 1), F(4, 5), G(11, 5), H(8, 1) 24. A(3, 9), B(8, 9), C(2, -3), D( -3, -3)
25.D(0, 0), E(2, 4), F(6, 4), G(6, 0)
26.K( -7, -2), L( -7, 6), M(1, 6), N(7, -2)
27.Explain Mathematical Ideas (1)(G) Ki
used geometry software to make
< > the figure
at the right. She< constructed
AB and a
>
point C not on AB
.
Then
she
constructed
< >
line k parallel to AB through point C. Next,
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.
C
D
k
B
A
The 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 = 1s(s − a)(s − b)(s − c), where s = 12 (a +hsm11gmse_1001_t09427.ai
b + c)
Use Heron’s Formula and a calculator to find the area of each triangle. Round your
answer to the nearest whole number.
28.a = 8 in., b = 9 in., c = 10 in.
29.a = 15 m, b = 17 m, c = 21 m
30.a. Use Heron’s Formula to find the area of this triangle.
b. Verify your answer to part (a) by using the formula A = 12 bh.
15 in.
9 in.
12 in.
TEXAS Test Practice
hsm11gmse_1001_t09184.ai
31.The lengths of the sides of a right triangle are 10 in., 24 in., and 26 in. What is the
area of the triangle?
A.
116 in.2 B.
120 in.2 C.
130 in.2 D.
156 in.2
32.In quadrilateral ABCD, AB ≅ BC ≅ CD ≅ DA. Which type of quadrilateral could
ABCD never be classified as?
F.
squareG.
rectangleH.
rhombusJ.
kite
33.Are the side lengths of △XYZ possible? Explain.
X
4
6
Z
Y
11
hsm11gmse_1001_t09429.ai
PearsonTEXAS.com
525
13-2 Areas of Trapezoids, Rhombuses,
and Kites
TEKS FOCUS
VOCABULARY
TEKS (11)(B) Determine the area of composite twodimensional figures comprised of a combination of triangles,
parallelograms, trapezoids, kites, regular polygons, or
sectors of circles to solve problems using appropriate units of
measure.
TEKS (1)(B) Use a problem–solving model that incorporates
analyzing given information, formulating a plan or strategy,
determining a solution, justifying the solution, and evaluating
the problem–solving process and the reasonableness of the
solution.
Additional TEKS (1)(A), (1)(F), (6)(D), (9)(B)
•Height of a trapezoid – The height of a trapezoid is
the perpendicular distance between the bases.
•Formulate – create with careful effort and purpose.
You can formulate a plan or strategy to solve a
problem.
•Strategy – a plan or method for solving a problem
•Reasonableness – the quality of being within the
realm of common sense or sound reasoning. The
reasonableness of a solution is whether or not the
solution makes sense.
ESSENTIAL UNDERSTANDING
• You can find the area of a trapezoid when you know its
height and the lengths of its bases.
• You can find the area of a rhombus or a kite when you
know the lengths of its diagonals.
Key Concept Area of a Trapezoid
The area of a trapezoid is half the product of the height and the sum of the bases.
A = 12h(b1 + b2)
b1
h
b2
Key Concept Area of a Rhombus or a Kite
hsm11gmse_1002_t09235.ai
The area of a rhombus or a kite is half the product of the lengths of its diagonals.
1
A = 2d1d2
d1
d2
d1
d2
526
Rhombus Kite
hsm11gmse_1002_t09243.ai
Lesson 13-2 Areas
of Trapezoids,hsm11gmse_1002_t14116.ai
Rhombuses, and Kites
Problem 1
TEKS Process Standard (1)(A)
Area of a Trapezoid
Which borders of
Nevada can you use
as the bases of a
trapezoid?
The two parallel sides of
Nevada form the bases
of a trapezoid.
Geography What is the approximate area of Nevada?
A = 12h(b1 + b2)
se the formula for area of
U
a trapezoid.
= 12(309)(205 + 511) S ubstitute 309 for h, 205 for
b1 , and 511 for b2 .
= 110,622
205 mi
309 mi
Reno
Carson City
511 mi
Simplify.
Las
Vegas
The area of Nevada is about 110,600 mi2.
Problem 2
hsm11gmse_1002_a09884
Nevada
10p x 13p
Finding Area Using a Right Triangle
What is the area of trapezoid PQRS?
How are the sides
related in a 30∙-60∙90∙ triangle?
The length of the
hypotenuse is 2 times the
length of the shorter leg,
and the longer leg is
13 times the length of
the shorter leg.
You can draw an altitude that divides the trapezoid into a rectangle
and a 30°-60°-90° triangle. Since the opposite sides of a rectangle
are congruent, the longer base of the trapezoid is divided into
segments of lengths 2 m and 5 m.
h = 213
longer leg = shorter leg
A = 12h(b1 + b2)
Use the trapezoid area formula.
= 12(213)(7 + 5) Substitute 213 for h, 7 for b1, and 5 for b2.
= 1213
Simplify.
The area of trapezoid PQRS is 1213
m2.
P
5m
60
S
R
Q
7m
# 13
S
1st proof
12.15.08
5m
R
hsm11gmse_1002_t09237.ai
h
60
P 2m
5m
Q
Problem 3
Finding the Area of a Kite
Do you need to know
the side lengths of
the kite to find its
area?
No. You only need the
lengths of the diagonals.
hsm11gmse_1002_t09241.ai
L
What is the area of kite KLMN?
Find the lengths of the two diagonals: KM = 2 + 5 = 7m and LN = 3 + 3 = 6m.
A = 12d1d2 2m
3m
K
Use the formula for area of a kite.
= 12(7)(6) Substitute 7 for d1 and 6 for d2 .
= 21
3m
5m
M
N
Simplify.
The area of kite KLMN is 21 m2.
hsm11gmse_1002_t09244.ai
PearsonTEXAS.com
527
Problem 4
Finding the Area of a Rhombus
How can you find the
length of AB?
AB is a leg of right
△ABC. You can use the
Pythagorean Theorem,
a2 + b2 = c 2, to find
its length.
Car Pooling The High Occupancy Vehicle (HOV)
lane is marked by a series of “diamonds,” or
rhombuses painted on the pavement. What is the
area of the HOV lane diamond shown at the right?
△ABC is a right triangle. Using the Pythagorean
Theorem, AB = 26.52 - 2.52 = 6. Since the
diagonals of a rhombus bisect each other, the
diagonals of the HOV lane diamond are 5 ft
and 12 ft.
A
6.5 ft
B 2.5 ft
A = 12 d1d2 Use the formula for area of a
rhombus.
= 12 (5)(12)
Substitute 5 for d1 and 12 for d2 .
= 30
Simplify.
C
The area of the HOV lane diamond is 30 ft2.
Problem 5
TEKS Process Standard (1)(B)
Finding the Area of a Composite Figure
Use a problem-solving model to find the area of the figure below.
6 yd
6 yd
3 yd
3 yd
4 yd
4 yd
10 yd
Analyze the Given Information
Using the definitions of a kite and a trapezoid, you can determine that this figure is
composed of a kite and two trapezoids. The diagonal of the kite is 10 yd long, and
the two trapezoids both have base lengths of 6 yd and 4 yd, and a height of 3 yd.
Formulate a Plan
To find the area of the composite figure, add the areas of each individual figure.
continued on next page ▶
528
Lesson 13-2 Areas of Trapezoids, Rhombuses, and Kites
Problem 5
continued
Determine and Justify the Solution
Find the area of each trapezoid.
A = 12 h(b1 + b2)
Use the formula for area of a trapezoid.
1
= 2 (3)(6 + 4)
Substitute 3 for h, 6 for b1 , and 4 for b2 .
= 15
Simplify.
Find the area of the kite. The length of the shorter diagonal is 2(6) - 2(4) = 4 yd.
A = 12 d1d2 Use the formula for area of a kite.
= 12 (4)(10)
Substitute 4 for d1 and 10 for d2 .
= 20
Simplify.
Find the total area. The total area is 15 + 15 + 20 = 50.
So the area of the composite figure is 50 yd2.
Evaluate the Problem-Solving Process
NLINE
HO
ME
RK
O
What should you do
if the answer doesn’t
check?
You should examine
the problem-solving
process to find mistakes
in your reasoning or your
calculations.
WO
Check your answer. You can divide the composite figure in a different way, find the area,
and compare your answers. You can divide the figure into a rectangle and an isosceles
triangle. The base of the rectangle is 12 yd and the height is 3 yd, so its area is 36 yd2. The
triangle has base 4 yd and height 7 yd, so its area is 14 yd2. 36 + 14 = 50, so the total area is
50 yd2. The answer checks.
Since the answer checks, the problem-solving model worked effectively in finding
the area of the composite figure.
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Find the area of quadrilateral QRST.
1.
y
For additional support when
completing your homework,
go to PearsonTEXAS.com.
R
2.
y
R
4
Q
3.
y R
2
2
x
2
T
S
-2
2 S 4
x
Q
O
-2
T
2
Q
S
2
4
x
2
T
4.
The border of Tennessee resembles a trapezoid with bases 340 mi and 440 mi and
height 110 mi. Estimate the area of Tennessee by finding the area of the trapezoid.
hsm11gmse_1002_t09301.ai
hsm11gmse_1002_t09302.ai
hsm11gmse_1002_t09300.ai
PearsonTEXAS.com
529
Find the area of each trapezoid. If your answer is not an integer, leave it in
simplest radical form.
5.
5 ft
6.
3 ft
7.
6m
8 ft
10 m
60
6 ft
8m
15 ft
Find the area of each kite.
8.
9. 2hsm11gmse_1002_t09284.ai
10.
m
2 in.
3m
hsm11gmse_1002_t09282.ai
8 in.
8 in.
8 in.
hsm11gmse_1002_t09283.ai
4m
6 ft
3m
4 ft
4 ft
Find the area of each rhombus.
11.
12.
f
20
hsm11gmse_1002_t09285.ai
t
30
ft
13.
10 in.
hsm11gmse_1002_t09290.ai
8 in.
hsm11gmse_1002_t09288.ai
6m
5m
Find the area of each trapezoid.
14.
21 in.
15.
hsm11gmse_1002_t09291.ai
16 in.
16.
24.3 cm
hsm11gmse_1002_t09293.ai hsm11gmse_1002_t09294.ai
9 ft
6 ft
8.5 cm
18 ft
9.7 cm
38 in.
17.Find the area of a trapezoid with bases 12 cm and 18 cm and height 10 cm.
hsm11gmse_1002_t09279.ai
18.Find the area of a trapezoid with bases 2 ft and 3 ft and height 13 ft.
hsm11gmse_1002_t09246.ai
19.Use a Problem-Solving Model (1)(B) Find the
area of the figure at the right. Use a problemsolving model by
• analyzing the given information
• formulating a plan or strategy
• determining a solution
• justifying the solution
• evaluating the problem-solving process
530
Lesson 13-2 Areas of Trapezoids, Rhombuses, and Kites
hsm11gmse_1002_t09281.ai
26 ft
13 ft
18 ft
24 ft
20.In trapezoid ABCD at the right, AB } DC. Find the area
of ABCD.
B
15 in.
A
20 in.
135
21.Analyze Mathematical Relationships (1)(F) One base of a
trapezoid is twice the other. The height is the average of the
two bases. The area is 324 cm2. Find the height and the
lengths of the bases.
30
D
C
22.Apply Mathematics (1)(A) Ty wants to paint one side of the skateboarding ramp
hsm11gmse_1002_t09310.ai
he built. The ramp is 4 m wide. Its surface is modeled by the equation
y = 0.25x2 .
Use the trapezoids and triangles shown to estimate the area to be painted.
y
1
y 0.25x2
x
2
1
O
1
23.Apply Mathematics (1)(A) The end of a gold bar
has the shape of a trapezoid with the measurements
shown. Find the area of the end.
hsm11gmse_1002_t09309.ai
24.a.Create Representations to Communicate
Mathematical Ideas (1)(E) Graph the lines
x = 0, x = 6, y = 0, and y = x + 4.
2
6.9 cm
4.4 cm
9.2 cm
b.What type of quadrilateral do the lines form?
c.Find the area of the quadrilateral.
TEXAS Test Practice
25.The area of a kite is 120 cm2. The length of one diagonal is 20 cm. What is the
length of the other diagonal?
A.12 cm
C.24 cm
B.20 cm
D.48 cm
26.△ABC ∼ △XYZ. AB = 6, BC = 3, and CA = 7. Which of the following are NOT
possible dimensions of △XYZ?
F.
XY = 3, YZ = 1.5, ZX = 3.5
G.XY = 9, YZ = 4.5, ZX = 10.5
H.XY = 10, YZ = 7, ZX = 11
J.XY = 18, YZ = 9, ZX = 21
27.Draw an angle. Construct a congruent angle and its bisector.
PearsonTEXAS.com
531
13-3 Areas of Regular Polygons
TEKS FOCUS
VOCABULARY
TEKS (11)(A) Apply the formula for the area of regular
polygons to solve problems using appropriate units of
measure.
•Apothem – An apothem is the perpendicular distance
from the center of a polygon to a side.
•Center of a regular polygon – The center of a regular
polygon is the center of a circle circumscribed about
the polygon.
TEKS (1)(F) Analyze mathematical relationships to connect
and communicate mathematical ideas.
Additional TEKS (1)(D), (9)(B)
•Radius of a regular polygon – A radius of a regular
polygon is the distance from the center of the
polygon to a vertex.
•Analyze – closely examine objects, ideas, or
relationships to learn more about their nature
ESSENTIAL UNDERSTANDING
The area of a regular polygon is related to the distance from the center to a side.
Postulate 13-2
If two figures are congruent, then their areas are equal.
Key Concept Area of a Regular Polygon
The area of a regular polygon is half the product of the apothem and
the perimeter.
A = 12ap
a
p
hsm11gmse_1003_t09350.ai
532
Lesson 13-3 Areas of Regular Polygons
Problem 1
TEKS Process Standard (1)(F)
Finding Angle Measures
The figure at the right is a regular pentagon with radii and an
apothem drawn. What is the measure of each numbered angle?
How do you know the
radii make isosceles
triangles?
Since the pentagon is
a regular polygon, the
radii are congruent. So
the triangle made by
two adjacent radii and a
side of the polygon is an
isosceles triangle.
360
m∠1 = 5 = 72
Divide 360 by the number of sides.
m∠2 = 12m∠1
T he apothem bisects the vertex angle of the
isosceles triangle formed by the radii.
90 + 36 + m∠3 = 180
The sum of the measures of the angles of a triangle is 180.
hsm11gmse_1003_t09347.ai
m∠3 = 54
2 1
= 12(72) = 36
3
m∠1 = 72, m∠2 = 36, and m∠3 = 54.
Problem 2
Finding the Area of a Regular Polygon
What is the area of the regular decagon shown below?
12.3 in.
What do you know
about the regular
decagon?
A decagon has 10 sides,
so n = 10. From the
diagram, you know
that the apothem a is
12.3 in., and the side
length s is 8 in.
8 in.
Step 1 Find the perimeter of the regular decagon.
p = ns
= 10(8)
= 80 in.
Use the formula for the perimeter of an n-gon.
hsm11gmse_1003_t09351.ai
Substitute
10 for n and 8 for s.
Step 2 Find the area of the regular decagon.
A = 12ap
= 12(12.3)(80)
= 492
Use the formula for the area of a regular polygon.
Substitute 12.3 for a and 80 for p.
The regular decagon has an area of 492 in.2.
PearsonTEXAS.com
533
Problem 3
TEKS Process Standard (1)(D)
Using Special Triangles to Find Area
STEM
Zoology A honeycomb is made up of regular hexagonal cells. The length of a side
of a cell is 3 mm. What is the area of a cell?
You know the length of a
side, which you can use to
find the perimeter.
The apothem
Draw a diagram to help find the
apothem. Then use the area formula
for a regular polygon.
Step 1 Find the apothem.
The radii form six 60° angles at the center, so you
can use a 30°-60°-90° triangle to find the apothem.
a = 1.513 longer leg = 13 # shorter leg
Step 2 Find the perimeter.
p = ns
Use the formula for the perimeter of an n-gon.
Substitute 6 for n and 3 for s.
= 6(3)
= 18 mm
30
534
3 mm
hsm11gmse_1003_t09353.ai
A = 12ap
Use the formula for the area of a regular polygon.
= 12(1.513) (18)
Substitute 1.513 for a and 18 for p.
≈ 23.3826859
Lesson 13-3 Areas of Regular Polygons
60
1.5 mm
Step 3 Find the area.
The area is about 23 mm2.
a
Use a calculator.
Problem 4
Finding the Area of a Composite Figure
The figure below is composed of two congruent regular hexagons and two triangles.
What is the area of the figure? Round your answer to the nearest square meter.
9m
Step 1 Find the area of one of the regular hexagons.
To find the area of a regular polygon, you need to know the apothem and the
perimeter. Use a 30°-60°-90° triangle to find the apothem. Since the hypotenuse
is 9 m long, the length of the apothem is 4.513 m. The perimeter of the hexagon
is 6 9 m, or 54 m.
#
A = 12ap
Use the formula for the area of a regular polygon.
= 12(4.513)(54) Substitute 4.5 13 for a and 54 for p.
= 121.5 13Simplify.
Step 2 Find the area of one of the triangles.
The measure of each angle of a regular hexagon is 120. So the measure
of an exterior angle is 180 - 120 = 60. Since two exterior angles of the
hexagons make up two angles of the triangle, the measure of all angles
of the triangle must be 60. Therefore, it is an equilateral triangle with side
length 9 m. Use another 30°-60°-90° triangle to find the height. Since the
hypotenuse is 9 m, the height is 4.513 m.
A = 12bh
Use the formula for the area of a triangle.
1
= 2(9)(4.5 13) Substitute 9 for b and 4.5 13 for h.
= 20.2513Simplify.
How do you know
that the two triangles
are congruent?
The triangles are both
equilateral, with side
lengths of 9 m. So they
are congruent.
Step 3 Find the area of the four figures combined.
A = 121.513 + 121.513 + 20.2513 + 20.2513
= 283.513
Simplify.
≈ 491.0364039
Use a calculator.
The area of the composite figure is about 491 m2.
PearsonTEXAS.com
535
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Each regular polygon has radii and apothem as shown. Find the measure
of each numbered angle.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
2.
3.
4
1
2
7
5
8
6
3
9
Find the area of each regular polygon with the given apothem a and side length s.
4.
pentagon, a = 24.3 cm, s = 35.3 cm
hsm11gmse_1003_t09363.ai
hsm11gmse_1003_t09364.ai
hsm11gmse_1003_t09367.ai
5.
octagon, a = 60.4 in., s = 50 in.
6.
nonagon, a = 27.5 in., s = 20 in.
7.
dodecagon, a = 26.1 cm, s = 14 cm
Find the area of each regular polygon. Round your answer to the nearest tenth.
8.
18 ft
9.
10.
8 in.
6m
11.Use Multiple Representations to Communicate Mathematical Ideas (1)(D) You are painting a mural of colored equilateral triangles. The radius of each triangle
is 12.7 in. What is the hsm11gmse_1003_t09369.ai
area of each triangle to the nearest square inch?
hsm11gmse_1003_t09368.ai
hsm11gmse_1003_t09370.ai
s
2
30
12.7 in.
s
Find the area of each regular polygon with the given radius or apothem. If your
answer is not an integer, leave it in simplest radical form.
13.hsm11gmse_1003_t09371.ai
14.
12.
4 in.
6 cm
536
8 V3 in.
hsm11gmse_1003_t09373.ai
hsm11gmse_1003_t09376.ai
hsm11gmse_1003_t09372.ai
Lesson 13-3 Areas of Regular Polygons
15.Apply Mathematics (1)(A) The gazebo in the photo is built in
the shape of a regular octagon. Each side is 8 ft long, and the
enclosed area is 310.4 ft2. What is the length of the apothem?
STEM
16.Apply Mathematics (1)(A) One of the smallest space satellites
ever developed has the shape of a pyramid. Each of the four faces
of the pyramid is an equilateral triangle with sides about 13 cm
long. What is the area of one equilateral triangular face of the
satellite? Round your answer to the nearest whole number.
17.A regular hexagon has perimeter 120 m. Find its area.
18.The area of a regular polygon is 36 in.2. Find the length of a
side if the polygon has the given number of sides. Round your
answer to the nearest tenth.
a.3
b.4
c.6
d.Select Techniques to Solve Problems (1)(C) Suppose the polygon is a
pentagon. What would you expect the length of a side to be? Explain.
19.A portion of a regular decagon has radii and an apothem drawn. Find the
measure of each numbered angle.
2
3
1
20.Explain Mathematical Ideas (1)(G) Explain why the radius of a regular
polygon is greater than the apothem.
Find the area of each composite
figure. Assume that all parts of figures shown
hsm11gmse_1003_t09377.ai
are regular polygons and that figures that are the same shape are congruent.
Leave your answer in simplest radical form.
22 in.
21.22.
23.
14 m
8 ft
Find the perimeter and area of each regular polygon. Round to the nearest
tenth, as necessary.
24.a square with vertices at ( -1, 0), (2, 3), (5, 0), and (2, -3)
25.a hexagon with two adjacent vertices at ( -2, 1) and (1, 2)
PearsonTEXAS.com
537
26.To find the area of an equilateral triangle, you can use the
formula A = 12bh or A = 12ap. A third way to find the area of
an equilateral triangle is to use the formula A = 14s2 13.
Verify the formula A = 14s2 13 in two ways, as follows:
a.Find the area of Figure 1 using the formula A = 12bh.
b.Find the area of Figure 2 using the formula A = 12ap.
s
s
s
2
s
2
Figure 1
27.For Problem 1, write a proof showing that the apothem
Proof bisects the vertex angle of an isosceles triangle formed by two radii.
Figure 2
28.Prove that the bisectors of the angles of a regular polygon are concurrent and that
Proof they are, in fact, radii of the polygon. (Hint: For regular n-gon hsm11gmse_1003_t09381.ai
ABCDE . . ., let P be
the intersection of the bisectors of ∠ABC and ∠BCD. Show that DP must be the
bisector of ∠CDE.)
y
29.Analyze Mathematical Relationships (1)(F) A regular
octagon with center at the origin and radius 4 is graphed in
the coordinate plane.
a.Since V2 lies on the line y = x, its x- and y-coordinates
are equal. Use the Distance Formula to find the
coordinates of V2 to the nearest tenth.
V2
2
2
O
2
V1 (4, 0)
x
2
b.Use the coordinates of V2 and the formula A = 12bh to
find the area of △V1OV2 to the nearest tenth.
c.Use your answer to part (b) to find the area of the
octagon to the nearest whole number.
hsm11gmse_1003_t09382.ai
TEXAS Test Practice
30.What is the area of a regular pentagon with an apothem of 25.1 mm and a
perimeter of 182 mm?
A.913.6 mm2
B.2284.1 mm2
C.3654.6 mm2
D.4568.2 mm2
31.What is the most precise name for a regular polygon with four right angles?
F.
square
G.parallelogram
H.trapezoid
J.rectangle
32.△ABC has coordinates A( -2, 4), B(3, 1), and C(0, -2). If you reflect △ABC
across the x-axis, what are the coordinates of the vertices of the image △A′B′C′?
A.A′(2, 4), B′( -3, 1), C′(0, -2)
C. A′(4, -2), B′(1, 3), C′( -2, 0)
B.A′( -2, -4), B′(3, -1), C′(0, 2)
D. A′(4, 2), B′(1, -3), C′( -2, 0)
33.An equilateral triangle on a coordinate grid has vertices at (0, 0) and (4, 0). What
are the possible locations of the third vertex? Explain.
538
Lesson 13-3 Areas of Regular Polygons
13-4 Perimeters and Areas of
Similar Figures
TEKS FOCUS
VOCABULARY
•Justify – explain with logical reasoning.
TEKS (10)(B) Determine and describe how changes in the linear
dimensions of a shape affect its perimeter, area, surface area, or volume,
including proportional and non-proportional dimensional change.
You can justify a mathematical
argument.
•Argument – a set of statements put
TEKS (1)(G) Display, explain, and justify mathematical ideas and
arguments using precise mathematical language in written or oral
communication.
forth to show the truth or falsehood
of a mathematical claim
Additional TEKS (1)(F), (11)(A)
ESSENTIAL UNDERSTANDING
You can use ratios to compare the perimeters and areas of similar figures.
Key Concept Changes in Dimension
Examples
A proportional dimensional change multiplies
every dimension by the same value. Under a
proportional dimensional change, the image of a
figure is similar to its preimage.
Under a nonproportional dimensional change,
the image of a figure is not similar to its preimage.
Two examples of nonproportional dimensional
changes are the following:
• Each dimension has the same constant value
added to it
• Each dimension is multiplied by a different value
a ∙ 1.5
a
b ∙ 1.5
b
a+2
b+2
a
b
a∙2
b∙3
a
b
Key Concept Perimeters and Areas of Similar Figures
a
If the scale factor of two similar figures is b , then
a
• the ratio of their perimeters is b
• the ratio of their areas is
a2
b2
PearsonTEXAS.com
539
Problem 1
TEKS Process Standard (1)(F)
Analyzing Proportional Dimensional Changes
How does multiplying each dimension of the
A 5 cm
isosceles trapezoid by a scale factor of 2 affect its
perimeter? How does multiplying each dimension
by a scale factor of 3 affect its perimeter?
5 cm
4 cm
5 cm
11 cm
Find the perimeter of the trapezoid.
P = 5 + 11 + 5 + 5 = 26
Find the dimensions and the perimeter of each scaled trapezoid. Compare the
new perimeter to the original perimeter of 26 cm.
Find the new
dimensions.
Scale Factor 2 Scale Factor 3
#
Find the new
perimeter.
#
#
b1 = 2 5 = 10
b2 = 2 11 = 22
leg = 2 5 = 10
h = 2 4 = 8
#
#
b1 = 3 5 = 15
b2 = 3 11 = 33
leg = 3 5 = 15
h = 3 4 = 12
#
#
#
P = 10 + 22 + 10 + 10P = 15 + 33 + 15 + 15
= 52
= 78
52 cm = 2
# 26 cm
78 cm = 3
# 26 cm
So, when each dimension is multiplied by 2, the perimeter is multiplied by 2.
When each dimension is multiplied by 3, the perimeter is multiplied by 3.
B How does multiplying each dimension of the isosceles trapezoid in Part A by a
scale factor of 2 affect its area? How does multiplying each dimension by a scale
factor of 3 affect its area?
Find the area of the trapezoid.
How does the ratio
of the areas appear
to be related to the
scale factors?
Since 2 2 = 4 and
3 3 = 9, the ratio of
the areas appears to be
the square of the scale
factors.
#
540
#
A = 12(4)(5 + 11) = 32
Use the dimensions you calculated in Part A to find the area of each scaled
trapezoid. Compare the new area to the original area of 32 cm2.
Scale Factor 2
A = 12 h(b1 + b2)
A=
= 12(8)(10 + 22) =
= 128
128 cm2 = 4
# 32 cm2
Scale Factor 3
1
2 h(b1 + b2)
1
2(12)(15 + 33)
= 288
288 cm2 = 9
# 32 cm2
So, when each dimension is multiplied by 2, the area is multiplied by 4. When each
dimension is multiplied by 3, the area is multiplied by 9.
Lesson 13-4 Perimeters and Areas of Similar Figures
Problem 2
Finding Ratios in Similar Figures
How do you find the
scale factor?
Write the ratio of
the lengths of two
corresponding sides.
The trapezoids at the right are similar. The ratio of the lengths of
corresponding sides is 69 , or 23 .
6m
A What is the ratio (smaller to larger) of the perimeters?
The ratio of the perimeters is the same as the ratio of
corresponding sides, which is 23 .
9m
B What is the ratio (smaller to larger) of the areas?
The ratio of the areas is the square of the ratio of corresponding
2
sides, which is 22 , or 49 .
3
Problem 3
hsm11gmse_1004_t09322.ai
Finding Areas Using Similar Figures
Can you eliminate
any answer choices
immediately?
Yes. Since the area of
the smaller pentagon
is 27.5 cm2, you know
that the area of the
larger pentagon must be
greater than that, so you
can eliminate choice A.
Multiple Choice The area of the smaller regular pentagon is
about 27.5 cm2. What is the best approximation for the area of the
larger regular pentagon?
11 cm2
69 cm2
172 cm2
4 cm
10 cm
275 cm2
Regular pentagons are similar because all angles measure 108 and all
sides in each pentagon are congruent. Here the ratio of corresponding side lengths
2
hsm11gmse_1004_t09323.ai
4
4
is 10
, or 25 . The ratio of the areas is 22 , or 25
.
5
27.5
4
25 = A Write a proportion using the ratio of the areas.
4A = 687.5
Cross Products Property
687.5
A= 4 Divide each side by 4.
A = 171.875
Simplify.
The area of the larger pentagon is about 172 cm2. The correct answer is C.
Problem 4
TEKS Process Standard (1)(G)
Applying Area Ratios
Do you need to know
the shapes of the
two plots of land?
No. As long as the plots
are similar, you can
compare their areas
using their scale factor.
Agriculture During the summer, a group of high school students cultivated a
plot of city land and harvested 13 bushels of vegetables that they donated to a
food pantry. Next summer, the city will let them use a larger, similar plot of land.
In the new plot, each dimension is 2.5 times the corresponding dimension of the
original plot. How many bushels can the students expect to harvest next year?
The ratio of the dimensions is 2.5 : 1. So the ratio of the areas is (2.5)2 : 12, or 6.25 : 1.
With 6.25 times as much land next year, the students can expect to harvest 6.25(13),
or about 81, bushels.
PearsonTEXAS.com
541
Problem 5
Finding Perimeter Ratios
The triangles shown below are similar. What is the scale factor? What is the ratio of
their perimeters?
Area 50 cm2
The areas of the two similar
triangles
a2 50
= b2 98
a2 25
= b2 49
a 5
= b 7
Area 98 cm2
The scale factor
hsm11gmse_1004_t15486
Write a proportion using the
ratios of the areas.
Use a2 : b2 for the ratio of the areas.
Simplify.
Take the positive square root of each side.
The ratio of the perimeters equals the scale factor 5 : 7.
Problem 6
Analyzing Nonproportional Dimension Changes
Are the rectangles
similar?
No. Since you cannot
apply the same scale
factor to the lengths of
each side of one of the
rectangles to get the
other, they are not similar.
The botany club plans to increase the size of a rectangular
garden by adding 8 ft to each dimension of the garden.
4 ft
A The botany club wants to put fencing around the
proposed garden. How many more feet of fencing will
the club need to buy for the proposed garden than it
would have bought for the current garden?
Find the perimeter of the current garden.
22 ft
4 ft
P = 2 # 22 + 2 # 16
= 76
Find the perimeter of the proposed garden.
P = 2(22 + 8) + 2(16 + 8)
= 108
16 ft
Find the difference of the two perimeters.
108 - 76 = 32
The botany club will need to buy 32 more feet of fencing for the proposed garden.
continued on next page ▶
542
Lesson 13-4 Perimeters and Areas of Similar Figures
Problem 6
continued
B The botany club wants to cover the proposed garden with a layer of mulch.
How much greater is the area of the proposed garden than the area of the
current garden?
Find the area of the current garden.
A = 22
# 16
= 352
Find the area of the proposed garden.
A = (22 + 8)(16 + 8)
= 720
Find the difference of the two areas.
720 - 352 = 368
NLINE
HO
ME
RK
O
So the area of the proposed garden is 368 yd2 greater than the area of the current
garden.
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
The figures in each pair are similar. Compare the first figure to the second.
Give the ratio of the perimeters and the ratio of the areas.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
2 in.
2.
3.
15 in.
4 in.
8 cm
25 in.
6 cm
Find the scale factor and the ratio of perimeters for each pair of similar figures.
4.
two regular octagons with areas 4 ft2 and 16 ft2
hsm11gmse_1004_t09327.ai
hsm11gmse_1004_t09325.ai
5.
two
trapezoids with areas 49 hsm11gmse_1004_t09330.ai
cm2 and 9 cm2
6.
two equilateral triangles with areas 1613 ft2 and 13 ft2
7.
two circles with areas 2p cm2 and 200p cm2
Analyze Mathematical Relationships (1)(F) Find the values of x and y when the
smaller triangle shown here has the given area.
8 cm
x
y
12 cm
8.
3 cm2
9.6 cm2
10.12 cm2
11.16 cm2
12.24 cm2
13.48 cm2
hsm11gmse_1004_t09335.ai
PearsonTEXAS.com
543
The figures in each pair are similar. The area of one figure is given. Find the
area of the other figure to the nearest whole number.
14.
15.
3 in.
12 m
6 in.
Area of smaller parallelogram = 6 in.2 16.
Area of larger trapezoid = 121 m2
17.
hsm11gmse_1004_t09331.ai
16 ft
18 m
hsm11gmse_1004_t09332.ai
12 ft
3m
11 m
Area of larger triangle = 105 ft2 Area of smaller hexagon = 23 m2
hsm11gmse_1004_t09333.ai
18.Apply Mathematics (1)(A) An embroidered placemat costs $3.95. An
embroidered tablecloth is similar to the placemat,
but four times as long and
hsm11gmse_1004_t09334.ai
four times as wide. How much would you expect to pay for the tablecloth?
19.The longer sides of a parallelogram are 5 m. The longer sides of a similar
parallelogram are 15 m. The area of the smaller parallelogram is 28 m2.
What is the area of the larger parallelogram?
STEM 20.Apply
Mathematics (1)(A) For some medical imaging, the scale of the image is
3 : 1. That means that if an image is 3 cm long, the corresponding length on the
person’s body is 1 cm. Find the actual area of a lesion if its image has area 2.7 cm2.
21.A rectangular pool and its scale drawing are similar, with a scale factor of
2.5 in. : 11.5 ft. If the dimensions of the drawing are 5.5 in. by 11 in., what is the
area of the bottom of the actual pool?
22.A rectangular driveway has a perimeter of 56 feet. If the length is increased by
4 feet, how is the perimeter affected? What is the new perimeter?
23.A postcard has side lengths s and t. Determine the changes in the area and
perimeter of the postcard if the length of s is tripled.
24.Explain Mathematical Ideas (1)(G) A reporter used the graphic below to show
that the number of houses with more than two televisions had doubled in the
past few years. Explain why this graphic is misleading.
Then
544
Now
Lesson 13-4 Perimeters and Areas of Similar Figures
STEM
25.a.Create Representations to Communicate Mathematical
Ideas (1)(E) A surveyor measured one side and two angles
of a field, as shown in the diagram. Use a ruler and a
protractor to draw a similar triangle.
b.Measure the sides and altitude of your triangle
and find its perimeter and area.
50
30
200 yd
c.Estimate the perimeter and area of the field.
26.Suppose the lengths of both bases of the isosceles trapezoid are halved.
Describe how the area of the trapezoid is affected.
27.a.Find the area of a regular hexagon with sides 2 cm long.
Leave your answer in simplest radical form.
b.Use your answer to part (a) and ratios to find the
areas of the regular hexagons shown at the right.
28.Justify Mathematical Arguments (1)(G) The enrollment at
an elementary school is going to increase from 200 students
to 395 students. A parents’ group is planning to increase the
100 ft-by-200 ft playground area to a larger area that is 200 ft
by 400 ft. What would you tell the parents’ group when they
ask your opinion about whether the new playground will be
large enough?
6 cm
7 cm
4 cm
HSM11GMSE_1004_a09815
12 cm
1st pass 12-19-08
Durke
3 cm
8 cm
hsm11gmse_1004_t09340.ai
29.The figure at the right is a scale drawing of a patio. The scale of the
drawing is 2 cm = 5 ft. What is the perimeter of the actual patio?
12 cm
30.A 3 in.-by-5 in. photograph is enlarged by a scale factor of 1 in. : 1.5 ft.
8 cm
a.Find the perimeter and the area of the enlarged photograph.
8 cm
b.Suppose the length and width of the enlarged photograph are
doubled. Describe how the perimeter and area are affected.
c.Suppose the length (the measure of the longer side) of the enlarged photograph
is doubled. Describe how the perimeter and area are affected.
TEXAS Test Practice
31.What is the value of x in the diagram at the right?
32.Two regular hexagons have sides in the ratio 3 : 5. The area of the
smaller hexagon is 81 m2. In square meters, what is the area of
the larger hexagon?
21
26
7
x
33.A trapezoid has base lengths of 9 in. and 4 in. and a height of 3 in.
What is the area of the trapezoid in square inches?
34.In quadrilateral ABCD, m∠A = 62, m∠B = 101, and m∠C = 42.
What is m∠D?
hsm11gmse_1004_t09536.ai
PearsonTEXAS.com
545
13-5 Trigonometry and Area
TEKS FOCUS
VOCABULARY
TEKS (9)(A) Determine the lengths of sides and measures of angles in a right triangle
by applying the trigonometric ratios sine, cosine, and tangent to solve problems.
TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil,
and technology as appropriate, and techniques, including mental math, estimation,
and number sense as appropriate, to solve problems.
•Number sense – the
understanding of what
numbers mean and how they
are related
Additional TEKS (1)(F), (11)(A)
ESSENTIAL UNDERSTANDING
• You can use trigonometry to find the area of a regular polygon
• You can use trigonometry to find the area of a triangle if
if you know the length of a side, a radius, or an apothem.
you know the lengths of two sides and the included angle.
Theorem 13-1 Area of a Triangle Given SAS
B
The area of a triangle is half the product of the lengths of two sides and the sine
of the included angle.
Area of △ABC = 12 bc (sin A)
a
c
A
b
C
For a proof of Theorem 13-1, see the Reference section on page 683.
Problem 1
TEKS Process Standard (1)(F)
hsm11gmse_1005_t10016.ai
Finding Area
What is the area of a regular nonagon with 10-cm sides?
What is the apothem
in the diagram?
The apothem is the
altitude of the isosceles
triangle. The apothem
bisects the central
angle and the side of
the polygon.
Draw a regular nonagon with center C. Draw CP and CR to form
isosceles △PCR. The measure of central ∠PCR is 360
9 , or 40.
The perimeter is 9 10, or 90 cm. Draw the apothem CS.
#
10 cm
C
m∠PCS = 12 m∠PCR = 20 and PS = 12 PR = 5 cm
Let a represent CS. Find a and substitute into the area formula.
5
tan 20° = a Use the tangent ratio.
5
a=
tan 20°
A = 12 ap
= 12 tan520°
#
# 90
≈ 618.1824194
Solve for a.
5
Substitute
for a and 90 for p.
tan 20°
Use a calculator.
P S
C
a
R
20
hsm11gmse_1005_t10011.ai
P 5 S
The area of the regular nonagon is about 618 cm2.
546
Lesson 13-5 Trigonometry and Area
hsm11gmse_1005_t10012.ai
Problem 2
TEKS Process Standard (1)(C)
Finding Area
Road Signs A stop sign is a regular octagon. The standard size has a 16.2-in. radius.
What is the area of the stop sign to the nearest square inch?
The radius and the number
of sides of the octagon
Use trigonometric ratios to find the
apothem and the length of a side
The apothem and the
length of a side
Step 1Let a represent the apothem. Use the cosine ratio to find a.
The measure of a central angle of the octagon is 360
8 , or 45.
So m∠C = 12 (45) = 22.5.
a
cos 22.5° = 16.2 16.2(cos 22.5°) = a
C
Use the cosine ratio.
22.5
Multiply each side by 16.2.
16.2
a
Step 2Let x represent AD. Use the sine ratio to find x.
x
sin 22.5° = 16.2 16.2(sin 22.5°) = x
Multiply each side by 16.2.
Step 3 Find the perimeter of the octagon.
The length of each side is 2x.
# length of a side
= 8 # 2x
= 8 # 2 # 16.2(sin 22.5°)
= 259.2(sin 22.5°)
Simplify.
A x D
Use the sine ratio.
hsm11gmse_1005_t10013.ai
p=8
Substitute for x.
7 4 2
. . . . . . .
Step 4 Substitute into the area formula.
A = 12ap
= 12 16.2(cos 22.5°)
#
# 259.2(sin 22.5°)
≈ 742.2924146
The area of the stop sign is about 742 in.2.
Substitute for a and p.
Use a calculator.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9
PearsonTEXAS.com
547
Problem 3
Finding Area
Which formula should
you use?
The diagram gives the
lengths of two sides
and the measure of the
included angle. Use the
formula for the area of a
triangle given SAS.
What is the area of the triangle?
Area = 12
= 12
# side length # side length # sine of included angle
# 12 # 21 # sin 48° Substitute.
≈ 93.63624801
12 cm
48
21 cm
Use a calculator.
The area of the triangle is about 94 cm2.
HO
ME
RK
O
hsm11gmse_1005_t10017.ai
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Select Tools to Solve Problems (1)(C) Find the area of each regular polygon. Round
your answers to the nearest tenth.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
octagon with side length 6 cm
2.decagon with side length 4 yd
3.
pentagon with radius 3 ft
4.nonagon with radius 7 in.
5.
dodecagon with radius 20 cm
6.20-gon with radius 2 mm
7.
18-gon with perimeter 72 mm
8.15-gon with perimeter 180 cm
Find the area of each triangle. Round your answers to the nearest tenth.
9.
11 m
12.
57
10.
6m
34 km
11. 104 m
12 ft
1
2
33
40
5 ft
226 m
13.
11 mm
1 ft
37
14.
28
76
hsm11gmse_1005_t10024.ai
1
hsm11gmse_1005_t10025.ai
hsm11gmse_1005_t10023.ai
2 ft
39 km
24 mm
2
15.Analyze Mathematical Relationships (1)(F) PQRST is a regular pentagon
with center O and radius 10 in. Find each measure. If necessary, round
your
answers to the nearest tenth.
hsm11gmse_1005_t10026.ai
hsm11gmse_1005_t10029.ai
hsm11gmse_1005_t10028.ai
P
T
X
S
O
Q
R
a.m∠POQ
b.m∠POX
c.OX
d.PQ
e.perimeter of PQRST
548
Lesson 13-5 Trigonometry and Area
f.area of PQRST
hsm11gmse_1005_t10027.ai
16.Apply Mathematics (1)(A) The Pentagon in Arlington, Virginia, is one of the
world’s largest office buildings. It is a regular pentagon, and the length of each of
its sides is 921 ft. What is the area of land that the Pentagon covers to the nearest
thousand square feet?
17.Apply Mathematics (1)(A) The surveyed lengths of two adjacent sides of a
triangular plot of land are 80 yd and 150 yd. The angle between the sides is 67°.
What is the area of the parcel of land to the nearest square yard?
Find the perimeter and area of each regular polygon to the nearest tenth.
18.
19.
4m
3 ft
20.
21.
1 mi
hsm11gmse_1005_t10030.ai
10 m
hsm11gmse_1005_t10031.ai
22.What is the area of the triangle shown below?
hsm11gmse_1005_t10032.ai
53
10 cm
hsm11gmse_1005_t10033.ai
79
8 cm
23.The central angle of a regular polygon is 10°. The perimeter of the polygon is
108 cm. What is the area of the polygon?
hsm11gmse_1005_t10034.ai
PearsonTEXAS.com
549
Regular polygons A and B are similar. Compare their areas.
24.The apothem of Pentagon A equals the radius of Pentagon B.
25.The length of a side of Hexagon A equals the radius of Hexagon B.
26.The radius of Octagon A equals the apothem of Octagon B.
27.The perimeter of Decagon A equals the length of a side of Decagon B.
28.Replacement glass for energy-efficient windows costs $5/ft2. About how
much will you pay for replacement glass for a regular hexagonal window
with a radius of 2 ft?
A.$10.39
B.$27.78
C.$45.98
D.$51.96
The polygons are regular polygons. Find the area of the shaded region.
29.
30.
8 cm
31.
6 ft
6 ft
4 in.
6 cm
32.Segments are drawn between the midpoints of consecutive sides of a regular
pentagon to form another regular pentagon. Find, to the nearest
hundredth, the
hsm11gmse_1005_t10036.ai
hsm11gmse_1005_t10035.ai
ratio of the area of the smaller pentagon to the area of the larger pentagon.
hsm11gmse_1005_t10037.ai
STEM 33.Apply Mathematics (1)(A) A surveyor wants to mark off a triangular parcel with
an area of 1 acre (1 acre = 43,560 ft2). One side of the triangle extends 300 ft along a
straight road. A second side extends at an angle of 65° from one end of the first side.
What is the length of the second side to the nearest foot?
TEXAS Test Practice
34.A regular polygon has a perimeter of 54 m and an apothem of 313 m. What is the
area of the polygon to the nearest tenth of a square meter?
35.The legs of a right triangle have lengths of 8 in. and 15 in. What is the length of the
hypotenuse in inches?
36.△PEN ≅ △LIV . If m∠P = 36 and m∠N = 82, what is m∠I ?
37.The perimeter of a parallelogram is 23.6 ft. If its length and width are doubled, what
is the perimeter of the parallelogram in feet?
38.The altitude to the hypotenuse of a right triangle divides the hypotenuse into
segments of lengths 8 and 10. What is the length of the shorter leg of the triangle?
550
Lesson 13-5 Trigonometry and Area
Topic 13 Review
TOPIC VOCABULARY
• altitude of a parallelogram, p. 520
• center of a regular polygon, p. 532
• height of a trapezoid, p. 526
• apothem, p. 532
• composite figure, p. 520
• height of a triangle, p. 520
• base of a parallelogram, p. 520
• height of a parallelogram, p. 520
• radius of a regular polygon, p. 532
• base of a triangle, p. 520
Check Your Understanding
Choose the vocabulary term that correctly completes the sentence.
1.A ? is a combination of two or more figures.
2.An ? is the perpendicular distance from the center of a polygon to a side.
3.The distance from the center to a vertex is the ? .
4.The length of the altitude of a parallelogram is also called the ? .
13-1 Areas of Parallelograms and Triangles
Quick Review
Exercises
You can find the area of a rectangle, a
parallelogram, or a triangle if you know
the base b and the height h.
The area of a rectangle or parallelogram
is A = bh.
Find the area of each figure.
h
5.
b
1
The area of a triangle is A = 2 bh.
6 ft
What is area of the parallelogram?
Use the area formula.
= (12)(8) = 96 Substitute and simplify.
The area of the parallelogram is 96 cm2.
10 in.
9 in.
4m
hsm11gmse_10cr_t10605.ai
7.
Example
A = bh
6.
5m
12 cm
8 cm
8.
10 ft
hsm11gmse_10cr_t10610.ai
16 ft
10 ft
hsm11gmse_10cr_t10609.ai
9.A right triangle has legs measuring 5 ft and 12 ft, and
hypotenuse measuring 13 ft. What is its area?
hsm11gmse_10cr_t10638 .ai
hsm11gmse_10cr_t10611.ai
hsm11gmse_10cr_t10637.ai
PearsonTEXAS.com
551
13-2 Areas of Trapezoids, Rhombuses, and Kites
Quick Review
Exercises
The area of a trapezoid is A = 12 h(b1 + b2).
The area of a rhombus or a kite is
d1
A = 12 d1d2 , where d1 and d2 are the
lengths of its diagonals.
Find the area of each figure. If necessary, leave your
answer in simplest radical form.
b2
The height of a trapezoid h is the
perpendicular distance between the bases,
b1 and b2.
h
10. b1
11 mm
8 ft
8 ft
d2
60
6
hsm11gmse_10cr_t10606.aimm
12. 10 ft
15 mm
13.
6.5 cm
10
8 cm
hsm11gmse_10cr_t10613.ai
ft
hsm11gmse_10cr_t10612.ai
.8
12
ft
Example
What is the area of the trapezoid?
A = 12 h(b1 + b2)
= 12 (8)(7 + 3) = 40
11.
Use the area formula.
3 cm
hsm11gmse_10cr_t10607.ai 10 cm
8 cm
6.5 cm
Substitute.
Simplify.
7 cm
14. A trapezoid has a height of 6 m. The length of one
base is three times the length
of the other base. The
hsm11gmse_10cr_t10640
.ai
sum of the base lengths is 18 m. What is the area of the
hsm11gmse_10cr_t10614.ai
trapezoid?
The area of the trapezoid is 40 cm2.
hsm11gmse_10cr_t10639 .ai
13-3 Areas of Regular Polygons
Quick Review
Exercises
The center of a regular polygon C
is the center of its circumscribed circle.
The radius r is the distance from the
center to a vertex. The apothem a is the
perpendicular distance from the center to
a side. The area of a regular polygon with
apothem a and perimeter p is A = 12 ap.
Find the area of each regular polygon. If your answer is
not an integer, leave it in simplest radical form.
C
r
a
15. 6 in.
16.
7 m
p
17. What is the area of a regular hexagon with a perimeter
of 240 cm?
Example
hsm11gmse_10cr_t10608.aihsm11gmse_10cr_t10641 .ai
18. What is the area of a square withhsm11gmse_10cr_t10642
radius 7.5 m?
.ai
What is the area of a hexagon with apothem 17.3 mm and
perimeter 120 mm?
Sketch each regular polygon with the given radius. Then
A = 12 ap
Use the area formula.
find its area to the nearest tenth.
= 12(17.3)(120) = 1038 Substitute and simplify.
The area of the hexagon is 1038 mm2.
19. triangle; radius 4 in.
20. square; radius 8 mm
21. hexagon; radius 7 cm
552
Topic 13 Review
13-4 Perimeters and Areas of Similar Figures
Exercises
Quick Review
a
If the scale factor of two similar figures is b , then the ratio of
a2
a
their perimeters is b , and the ratio of their areas is 2 .
b
For each pair of similar figures, find the ratio of the area
of the first figure to the area of the second.
22. 8
12
23.
Example
If the ratio of the areas of two similar figures is 49 ,
what is the ratio of their perimeters?
6
Find the scale factor.
14 2
= 19 3
24. Take the square root of the ratio of areas.
4
25.
3
hsm11gmse_10cr_t10615.ai
6
7 mm
14 mm
hsm11gmse_10cr_t10616.ai
The ratio of the perimeters is the same as the ratio of
corresponding sides, 23 .
26. If the ratio of the areas of two similar hexagons is 8 : 25,
what is the ratio of their perimeters?
hsm11gmse_10cr_t10631.ai
hsm11gmse_10cr_t10617.ai
13-5 Trigonometry and Area
Quick Review
Exercises
You can use trigonometry to find the areas of regular
polygons. You can also use trigonometry to find the area of
a triangle when you know the lengths of two sides and the
measure of the included angle.
Find the area of each polygon. Round your answers to the
nearest tenth.
Area of a △
= 12 side length
#
27. regular decagon with radius 5 ft
28. regular pentagon with apothem 8 cm
# side length # sine of included angle
29. regular hexagon with apothem 6 in.
30. regular quadrilateral with radius 2 m
Example
31. regular octagon with apothem 10 ft
What is the area of △XYZ?
Area = 12
= 12
X
# XY # XZ # sin X
# 15 # 13 # sin 65°
≈ 88.36500924
The area of △XYZ is approximately
88 ft2.
15 ft
Y
65
32. regular heptagon with radius 3 ft
13 ft
Z
33. 34.
15 cm
12 m
45
19 cm
78
12 m
hsm11gmse_10cr_t10618.ai
hsm11gmse_10cr_t10619.ai
hsm11gmse_10cr_t10620.ai
PearsonTEXAS.com
553
Topic 13 TEKS Cumulative Practice
Multiple Choice
5.Every triangle in the figure at the right is
an equilateral triangle. What is the total
area of the shaded triangles?
Read each question. Then write the letter of the correct
answer on your paper.
s2 13
s2 13
s
6.Which of the following could be the side lengths of
a right triangle?
B.
25 m2D.
513 m2
2.If △CAT is rotated 90° around
vertex C, what are the coordinates
of A′?
4
F.
4.1, 6.2, 7.3H.
3.2, 5.4, 6.2
y
C
G.
40, 60, 72J.
33, 56, 65
2
F.
(1, 5)
T
A
O
2
2
x
4
2
H.
( -1, 3)
hsm11gmse_10cu_t10743.ai
7.The vertices of △ART are A(1, -2), R(5, -1), and
T(4, -4). If T65, -27 (△ART) = A′R′T′, what are the
coordinates of the image triangle A′R′T′?
A.
A′(5, -1), R′(4, -4), T′(1, -2)
B.
A′(6, -4), R′(10, -3), T′(9, -6)
J.
(3, 5)
3.Which transformation can you use to justify that △ABC
is congruent to △PQR?
A
4
2
D.
A′( -1, 3), R′(3, 4), T′(2, 1)
B
R
6 ft
x
4
4 2
2
P
C.
A′(3, 3), R′(7, 4), T′(6, 1)
Bhsm11gmse_10cu_t10740.ai
8.In the figure below, what is the length of AD?
y
O
C
4
Q
A.
a reflection across the y-axis
B.
a reflection across the x-axis
C.
a rotation 180° clockwise around the origin
hsm11gmse_10cu_t10741.ai
D.
a translation 5 units left and 6 units down
4.Your friend is 5 ft 6 in. tall. When your friend’s shadow is
6 ft long, the shadow of a nearby sculpture is 30 ft long.
What is the height of the sculpture to the nearest tenth?
F.
32.7 ftH.
27.5 ft
G.
28 ftJ.
25 ft
554
s2 13
B.12 D.
6
A.
2513 m2 C.
1013 m2
G.
(3, 3)
s2 13
A.36 C.
9
1.What is the exact area of an equilateral triangle with
sides of length 10 m?
Topic 13 TEKS Cumulative Practice
A
D 4 ft C
F.
2 113 ftH.
10 ft
G.
9 ftJ.
3 113 ft
hsm11gmse_10cu_t10745.ai
9.For what value
of x are the two triangles similar?
5x 2
2
3
3x 2
6
4
1
A.
4
2 C.
B.
2D.
6
hsm11gmse_09cu_t08650.ai
Gridded Response
10. Find the area, in square meters, of the figure.
18. A photographic negative is 3 cm by 2 cm. A similar
print from the negative is 9 cm long on its shorter side.
What is the length, in centimeters, of the longer side?
19. What is the geometric mean of 10 and 15?
5m
3m
8m
Constructed Response
5m
11. Find the area, in square millimeters, of the regular
polygon. Round to the nearest tenth.
10 mm
12. A triangle has a perimeter of 81 in. If you divide the
length of each side by 3, what is the perimeter, in
inches, of the new triangle?
13. The diagonal of a rectangular patio makes a 70° angle
with a side of the patio that is 60 ft long. What is the
area, to the nearest square foot, of the patio?
14. What is the area, in square feet, of the unshaded part of
the rectangle below?
250 ft
120 ft
50 ft
100 ft
15. You are making a scale model of a building. The front of
the actual building is 60 ft wide and 100 ft tall. The front
of your model is 3 ft wide and 5 ft tall. What is the scale
factor ofhsm11gmse_10cu_t10747.ai
the reduction?
16. A square has an area of 225 cm2. If you double
the length of each side, what is the area, in square
centimeters, of the new square?
17. A meter stick perpendicular to the ground casts a 1.5-m
shadow. At the same time, a telephone pole casts a shadow
that is 9 m. How tall, in meters, is the telephone pole?
20. The coordinates of △ABC are A(2, 3), B(10, 9), and
C(10, -3). Is △ABC an equilateral triangle? Explain.
21. Draw a right triangle. Then construct a second
triangle congruent to the first. Show your steps.
22. One diagonal of a parallelogram has endpoints at
P( -2, 5) and R(1, -1). The other diagonal has
endpoints at Q(1, 5) and S( -2, -1). What type of
parallelogram is PQRS? Explain.
23. A right triangle has height 7 cm and base 4 cm. Find
its area using the formulas A = 12 bh and
A = 12 ab(sin C). Are the results the same? Explain.
24. Can a regular polygon have an apothem and a radius
of the same length? Explain.
25. The coordinates of the vertices of isosceles trapezoid
ABCD are A(0, 0), B(6, 8), C(10, 8), and D(16, 0). The
coordinates of the vertices of isosceles trapezoid
AFGH are A(0, 0), F(3, 4), G(5, 4), and H(8, 0). Are the
two trapezoids similar? Justify your answer.
26. At a campground, the 50-yd path from your campsite
to the information center forms a right angle with
the path from the information center to the lake. The
information center is located 30 yd from the
bathhouse. How far is your campsite from the lake?
Show your work.
Information center
50 yd
Your campsite
30 yd
x
Bathhouse y
Lake
hsm11gmse_07cu_t05666
PearsonTEXAS.com
555