Download Areas of Parallelograms and Triangles

Areas of Parallelograms
and Triangles
10-1
Vocabulary
Review
The diagram below shows the different types of parallelograms.
Parallelogram
Rectangle
Rhombus
Square
Underline the correct word to complete each sentence.
2. All parallelograms have opposite sides parallel / perpendicular .
3. Some parallelograms are trapezoids / rectangles .
Vocabulary Builder
area (noun)
EHR
ee uh
Definition: Area is the number of square units needed to cover a given surface.
Main Idea: You can find the area of a parallelogram or a triangle when you know the
length of its base and its height.
Use Your Vocabulary
Find the area of each figure.
4.
5.
18 square units
Chapter 10
6.
4.5 square units
18 square units
250
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1. All parallelograms are quadrilaterals / rectangles .
Theorems 10-1 and 10-2 Area of a Rectangle and a Parallelogram
Theorem 10-1 Area of a Rectangle
The area of a rectangle is the product of its
base and height.
Theorem 10-2 Area of a Parallelogram
The area of a parallelogram is the product of a
base and the corresponding height.
A 5 bh
A 5 bh
h
h
b
b
7. Explain how finding the area of a parallelogram and finding the area of a rectangle
are alike. Explanations may vary. Sample:
For each figure, you find the product of the base and
_______________________________________________________________________
its corresponding height.
_______________________________________________________________________
Problem 1 Finding the Area of a Parallelogram
Got It? What is the area of a parallelogram with base length 12 m and
height 9 m?
8. Label the parallelogram at the right.
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9. Find the area.
A 5 bh
9m
Write the formula.
5 12 ( 9 )
Substitute.
5 108
Simplify.
12 m
10. The area of the parallelogram is 108 m2 .
Problem 2 Finding a Missing Dimension
Got It? A parallelogram has sides 15 cm and 18 cm. The height corresponding to a
15-cm base is 9 cm. What is the height corresponding to an 18-cm base?
11. Label the parallelogram at the right.
Let h represent the height corresponding
to the 18-cm base.
h
9 cm
12. Find the area.
A 5 bh
5 15 ? 9
5 135
18 cm
15 cm
13. The area of the parallelogram is 135 cm2 .
251
Lesson 10-1
14. Use the area of the parallelogram to find the height corresponding to an
18-cm base.
A 5 bh
Write the formula.
135 5 ( 18 )h
( 18 )h
135 5
Substitute.
Divide each side by the length of the base.
18
18
7.5 5 h
Simplify.
15. The height corresponding to an 18-cm base is 7.5 cm.
Theorem 10-3 Area of a Triangle
The area of a triangle is half the product of a base and
the corresponding height.
h
b
A 5 12 bh
16. Explain how finding the area of a triangle is different from finding the
area of a rectangle. Explanations may vary. Sample:
For a triangle, find half the product of the base and height. For a
____________________________________________________________________________
Problem 3 Finding the Area of a Triangle
Got It? What is the area of the triangle?
5 in.
17. Circle the formula you can use to find the area of the triangle.
A 5 12 bh
18. Convert the lengths of the base and the hypotenuse to inches.
A 5 bh
base
hypotenuse
1 ft 5 12 in.
1 ft 1 in. 5 13 in.
19. Find the area of the triangle.
A 5 12bh
5 12(12)(5)
5 12(60)
5 30
20. The area of the triangle is 30 in.2 .
Chapter 10
252
1 ft 1 in.
1 ft
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rectangle, find the product of the base and height.
____________________________________________________________________________
Problem 4 Finding the Area of an Irregular Figure
Got It? Reasoning Suppose the base lengths of the square and triangle in the figure
are doubled to 12 in., but the height of each polygon remains the same. How is the
area of the figure affected?
8 in.
21. Complete to find the area of each irregular figure.
Area of Original Irregular Figure
6 in.
Area of New Irregular Figure
A 5 6(6) 1 12(6)(8)
1
A 5 (2)(6)(6) 1 2(2)(6)(8)
5 (2)(36) 1 (2) ( 24 )
5 36 1 24
5 60
5 (2)( 36 1 24 ) 5 (2) ( 60 ) 5 120
22. How is the area affected?
The
area is doubled.
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
D
~ABCD is divided into two triangles along diagonal AC. If you know the
area of the parallelogram, how do you find the area of kABC?
C
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Write T for true or F for false.
T
23. Since AC is a diagonal of ~ABCD, nABC is congruent to nCDA.
F
24. The area of nABC is greater than the area nCDA.
T
25. The area of nABC is half the area of ~ABCD.
A
B
26. If you know the area of the parallelogram, how do you find the area of nABC ?
Divide the area of the parallelogram by 2.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
base of a parallelogram
height of a parallelogram
base of a triangle
height of a triangle
Rate how well you can find the area of parallelograms and triangles.
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253
Lesson 10-1
Areas of Trapezoids,
Rhombuses, and Kites
10-2
Vocabulary
Review
1. Is a rhombus a parallelogram?
Yes / No
2. Are all rhombuses squares?
Yes / No
3. Are all squares rhombuses?
Yes / No
4. Cross out the figure that is NOT a rhombus.
10
7
7
10
5
5
5
5
3
3
3
3
kite (noun) kyt
Definition: A kite is a quadrilateral with two pairs of congruent adjacent sides.
Main Idea: You can find the area of a kite when you know the lengths of its
diagonals.
Word Origin: The name for this quadrilateral is taken from the name of the flying
toy that it looks like.
Use Your Vocabulary
5. Circle the kite.
4
2
2
4
4
2
2
4
2
4
2
4
6. The figure at the right is a kite. What is the value of x? Explain.
3
5
3
x
5; a kite has two pairs of congruent adjacent sides.
__________________________________________________________________
Chapter 10
254
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Vocabulary Builder
Theorem 10-4 Area of a Trapezoid
b1
The area of a trapezoid is half the product of the height
and the sum of the bases.
h
A 5 12h(b1 1 b2)
Underline the correct word to complete each sentence.
b2
7. The bases of a trapezoid are parallel / perpendicular .
8. The height / width of a trapezoid is the perpendicular distance between the bases.
Problem 1 Area of a Trapezoid
Got It? What is the area of a trapezoid with height 7 cm and bases
12 cm and 15 cm?
9. Use the justifications below to find the area of the trapezoid.
1
A 5 2 h(b1 1 b2)
1
5 2 ( 7 )( 12 1 15)
Substitute.
5 12 ( 7 )( 27 )
Add.
5
Simplify.
94.5
10. The area of the trapezoid is
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Use the formula for area of a trapezoid.
94.5
cm2 .
Problem 2 Finding Area Using a Right Triangle
Got It? Suppose h decreases in trapezoid PQRS so that mlP 5 45 while angles
S
5m
R
5m
Q
R and Q and the bases stay the same. What is the area of trapezoid PQRS?
11. If m/P 5 45, is the triangle still a 308-608-908 triangle?
Yes / No
12. Is the triangle a 458-458-908 triangle?
Yes / No
13. Are the legs of a 458-458-908 triangle congruent?
Yes / No
h
60
P 2m
14. The height of the triangle is 2 m.
15. The area is found below. Write a justification for each step.
A 5 12 h(b1 1 b2)
Use the trapezoid area formula.
5 12(2)(5 1 7)
Substitute.
5 12(2)(12)
Add.
5 12
Simplify.
16. The area of trapezoid PQRS is 12 m2 .
255
Lesson 10-2
Theorem 10-5 Area of a Rhombus or a Kite
The area of a rhombus or a kite is half the product of the
lengths of its diagonals.
d2
d1
d1
1
d2
A 5 2(d1d2)
Rhombus
Kite
17. Describe one way that finding the area of rhombus or a kite is different from
finding the area of a trapezoid.
Answers may vary. Sample: Instead of using the height and the
_______________________________________________________________________
lengths of the bases, you use the lengths of the diagonals.
_______________________________________________________________________
_______________________________________________________________________
18. Find the lengths of the diagonals of the kite and the rhombus below.
3m
2m
4m
4m
2m
lengths of the diagonals of the kite:
6 m and 4
4m
3m
lengths of the diagonals of the rhombus:
8 m and 6 m
m
Problem 3 Finding the Area of a Kite
Got It? What is the area of a kite with diagonals that are 12 in. and 9 in. long?
19. Error Analysis Below is one student’s solution. What error did the student make?
A = 1 (12 + 9)
2
= 1 (21)
2
= 10.5
Answers may vary. Sample: The student
_____________________________________________________
added the lengths of the diagonals instead
_____________________________________________________
of multiplying them.
_____________________________________________________
20. Find the area of the kite.
A 5 12(12)(9)
5 12(108)
5 54
21. The area of the kite is 54 in.2 .
Chapter 10
256
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2m
Problem 4 Finding the Area of a Rhombus
Got It? A rhombus has sides 10 cm long. If the longer diagonal is 16 cm, what is the
area of the rhombus?
Underline the correct words to complete the sentence.
10 cm
22. The diagonals of a rhombus bisect each other / side and
are parallel / perpendicular .
8 cm
23. Label the rhombus at the right.
24. The shorter diagonal is x
x cm
1
10 cm
10 cm
8 cm
10 cm
x , or 2x .
25. Use the Pythagorean Theorem to
find the value of x.
26. Find the area of the rhombus.
A 5 12(12)(16)
5 12 (192)
x 2 1 82 5 102
x 2 1 64 5 100
x 2 5 36
x56
5 96
27. The area of the rhombus is 96 cm 2 .
Lesson Check • Do you UNDERSTAND?
Reasoning Do you need to know the lengths of the sides to find the area of a kite? Explain.
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28. Cross out the length you do NOT need to find the area of each triangle in a kite.
each leg
hypotenuse
29. Now answer the question. Explanations may vary. Sample:
No. Each side of a kite is a hypotenuse of a right triangle.
_______________________________________________________________________
You need only the lengths of the legs to find the areas
_______________________________________________________________________
of the four right triangles that form the kite.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
kite
height of a trapezoid
Rate how well you can find the area of a trapezoid, rhombus, or kite.
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257
Lesson 10-2
Areas of Regular Polygons
10-3
Vocabulary
Review
Write T for true or F for false.
T
1. In a regular polygon, all sides are congruent.
F
2. In a regular polygon, all angles are acute.
3. Cross out the figure that is NOT a regular polygon.
5
4
3
3
4
5
5
5
5
4
2
2
3
4
5
apothem (noun)
AP
uh them
Related Words: center, regular polygon
Definition: The apothem is the perpendicular distance from the center of a regular
polygon to one of its sides.
Use Your Vocabulary
4. Underline the correct word to complete the statement.
In a regular polygon, the apothem is the perpendicular distance from
the center to a(n) angle / side .
5. Label the regular polygon below using apothem, center, or side.
center
apothem
side
Chapter 10
258
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Vocabulary Builder
Problem 1 Finding Angle Measures
Got It? At the right, a portion of a regular octagon has radii and an
1
apothem drawn. What is the measure of each numbered angle?
2
3
6. A regular octagon has 8 sides.
7. Circle the type of triangles formed by the radii of the regular octagon.
equilateral
isosceles
right
8. Use the justifications below to find the measure of each numbered angle.
m/1 5 360 5
8
1
2
m/2 5
5 12 (
45
Divide 360 by the number of sides.
(m/1)
45
)5
22.5
90 1 m/2 1 m/3 5
90 1
The apothem bisects the vertex angle of the
triangle formed by the radii.
180
Triangle Angle-Sum Theorem
22.5
1 m/3 5
180
Substitute.
112.5
1 m/3 5
180
Simplify.
m/3 5
67.5
Subtraction Property of Equality
9. Write the measure of each numbered angle.
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m/1 5
m/2 5
45
m/3 5
22.5
67.5
Postulate 10-1 and Theorem 10-6
Postulate 10-1 If two figures are congruent, then their areas are equal.
B
The isosceles triangles in the regular hexagon at the right are congruent.
Complete each statement.
A
10. If the area of nAOB is 24 in.2 , then the area of nBOC is 24 in.2 .
11. If the area of nBOC is 8
cm2 , then the area of
nAOC is 16
C
O
cm2 .
Theorem 10-6 Area of a Regular Polygon
The area of a regular polygon is half the product of the
apothem and the perimeter.
a
p
1
A 5 2 ap
Complete.
12. apothem: 10
perimeter: 80
area: 12 (10) ?
13. apothem: 5
perimeter: 30!3
area: 12 ?
14. apothem: 5!3
perimeter: 60
area:
259
80
? 30 !3
5
1
2
?
5 !3 ?
60
Lesson 10-3
Problem 2
Finding the Area of a Regular Polygon
Got It? What is the area of a regular pentagon with an 8-cm apothem
and 11.6-cm sides?
15. Label the regular pentagon with the lengths of the apothem and the sides.
11.6 cm
16. Use the justifications below to find the perimeter.
p 5 ns
5
5
Use the formula for the perimeter of an n-gon.
(11.6)
5 58
8 cm
Substitute for n and for s.
Simplify.
17. Use the justifications below to find the area.
A 5 12 ap
5 12 ?
Use the formula for the area of a regular polygon.
8
? 58
Substitute for a and for p.
5 232
Simplify.
18. The regular pentagon has an area of 232 cm2 .
Problem 3 Using Special Triangles to Find Area
Got It? The side of a regular hexagon is 16 ft. What is the area of the hexagon?
Round your answer to the nearest square foot.
Need
The length of the
apothem and the
perimeter
Know
I know that the length of
each side of the regular
hexagon is 16 ft.
Plan
Draw a diagram to help
find the length of the
apothem.
Then use the perimeter
and area formulas.
Use the diagram at the right.
20. Label the diagram.
21. Circle the relationship you can use to find the length of the apothem.
hypotenuse 5 2 ? shorter leg
longer leg 5 !3 ? shorter leg
22. Complete.
length of shorter leg 5
8 ft
8
ft
length of longer leg (apothem) 5 8 !3 ft
23. Use the formula p 5 ns to find the perimeter of the hexagon.
p 5 ns
5 6(16)
5 96
Chapter 10
30î
260
60î
a
16 ft
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19. Use the information in the problem to complete the problem-solving model below.
24. Now use the perimeter and the formula A 5 12 ap to find the area of the hexagon.
A 5 12ap
5 12(8 !3)(96)
5 665.1075101
25. To the nearest square foot, the area of the hexagon is 665 ft2 .
Lesson Check • Do you UNDERSTAND?
What is the relationship between the side length and the apothem in each figure?
square
45
regular hexagon
30
a
s
equilateral triangle
30í
a
a
s
s
26. The radius and apothem form what type of triangle in each figure?
square
45 8 - 45 8 - 90 8 triangle
regular hexagon
30 8 - 60 8 - 90 8 triangle
equilateral triangle
30 8 - 60 8 - 90 8 triangle
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27. Complete to show the relationship between the side length and the apothem.
square
regular hexagon
leg 5 leg
a5
1
2
equilateral triangle
longer leg 5 !3 ? shorter leg
a 5 !3 ?
s
a5
!3
2
1
2
longer leg 5 !3 ? shorter leg
1
2 s 5 !3 ?
s
a
s 5 2 !3 a
s
Math Success
Check off the vocabulary words that you understand.
radius of a regular polygon
apothem
Rate how well you can find the area of a regular polygon.
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261
Lesson 10-3
Perimeters and Areas
of Similar Figures
10-4
Vocabulary
Review
1. What does it mean when two figures are similar?
Answers may vary. Sample: The figures have the same shape.
_______________________________________________________________________
2. Are the corresponding angles of similar figures always congruent?
Yes / No
3. Are the corresponding sides of similar figures always proportional?
Yes / No
4. Circle the pairs of similar figures.
3
2
2
6
6
3
4
4
6
4
8
8
6
6
8
8
6
8
radius (noun)
RAY
dee us (plural radii)
Related Words: apothem, center
Definition: The radius of a regular polygon is the distance from the center
to a vertex.
Main Idea: The radii of a regular polygon divide the polygon into
congruent triangles.
Use Your Vocabulary
5. Cross out the segment that is NOT a radius of regular pentagon ABCDE.
OA
OD
OB
OE
OC
OF
C
B
Underline the correct word(s) to complete each sentence.
6. The radii of a regular polygon are / are not congruent.
7. The triangles formed by the radii and sides of regular pentagon ABCDE
are / are not congruent.
Chapter 10
262
D
O
A
F
E
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Vocabulary Builder
Theorem 10-7 Perimeters and Areas of Similar Figures
a
If the scale factor of two similar figures is b , then
a
(1) the ratio of their perimeters is b and
(2) the ratio of their areas is
a2
.
b2
8. The name for the ratio of the length of one side of a figure to the
length of the corresponding side of a similar figure is the 9.
scale factor
1
9. If the scale factor of two figures is 12 , then the ratio of their perimeters is
.
2
3
3
10. If the scale factor of two figures is x , then the ratio of their perimeters is
3
11. If the scale factor of two figures is 5 , then the ratio of their areas is
12. If the scale factor of two figures is x1 then the ratio of their areas is
3
5
1
x
x
.
2
2.
2
2.
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Problem 1 Finding Ratios in Similar Figures
Got It? Two similar polygons have corresponding sides in the ratio 5 : 7. What is
the ratio (larger to smaller) of their perimeters? What is the ratio (larger to smaller)
of their areas?
13. Circle the similar polygons that have corresponding sides in the ratio 5 : 7.
14
8
5
10
14
10
Underline the correct word to complete each sentence.
14. In similar figures, the ratio of the areas / perimeters equals the ratio of
corresponding sides.
15. In similar figures, the ratio of the areas / perimeters equals the ratio of the squares
of corresponding sides.
16. Complete.
ratio (larger to smaller)
of corresponding sides
ratio (larger to smaller)
of perimeters
ratio (larger to smaller)
of areas
7
7
7
5
5
5
263
2
2
5
49
25
Lesson 10-4
Problem 2
Finding Areas Using Similar Figures
Got It? The scale factor of two similar parallelograms is 34 . The area of the larger
parallelogram is 96 in.2.What is the area of the smaller parallelogram?
Write T for true or F for false.
3
F
17. The ratio of the areas is 4 .
T
18. The ratio of the areas is 16 .
9
19. Use the justifications below to find the area A of the smaller parallelogram.
9
16
A
5 96
Write a proportion.
16A 5 (96) 9
Cross Products Property
16A 5 864
Multiply.
16A 5 864
Divide each side by 16 .
16
16
A 5 54
Simplify.
20. The area of the smaller parallelogram is 54 in.2 .
Problem 3 Applying Area Ratios
Got It? The scale factor of the dimensions of two similar pieces of window glass is
21. Use the information in the problem to complete the reasoning model below.
Write
Think
The ratio of areas is the square of the
Ratio of areas â32 : 52
scale factor.
â 9 : 25
I can use a proportion to find the cost c of
9
â 2.50
c
the larger piece to the nearest hundredth.
25
9
Ƃ c â2.50 Ƃ
9
Ƃ c â 62.5
9
9
c
62.5
â
9
c Ƽ 6.94
22. The larger piece of glass should cost about $ 6.94 .
Chapter 10
264
25
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3 : 5. The smaller piece costs $2.50. How much should the larger piece cost?
Problem 4 Finding Perimeter Ratios
Got It? The areas of two similar rectangles are 1875 ft2 and 135 ft2 .
What is the ratio of their perimeters?
23. The scale factor is found below. Use one of the reasons listed in the blue
box to justify each step.
a2
135
5 1875
b2
Write a proportion.
a2
9
5 125
b2
Simplify.
Rationalize the
denominator.
Simplify.
Simplify.
a
3
5
b
5 !5
Take the positive square root of each side.
!5
a
3
5
?
b
5 !5 !5
Rationalize the denominator.
3 !5
a
5 25
b
Simplify.
24. The ratio of the perimeters equals the scale factor
Take the positive
square root of
each side.
Write a
proportion.
3 !5
:
25
.
Lesson Check • Do you UNDERSTAND?
Reasoning The area of one rectangle is twice the area of another. What is the ratio of
their perimeters? How do you know?
25. Let x and y be the sides of the smaller rectangle. Complete.
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area of smaller rectangle
area of larger rectangle
ratio of larger to smaller areas
2xy
2 xy
xy
:
xy
26. Find the square root of the ratio of larger to smaller areas to find the scale factor.
!2xy
!xy
!2
2xy
5 Å xy 5 Å21 5 1
27. The ratio of perimeters is
!2
:
1
because the scale factor is
!2
:
1
.
Math Success
Check off the vocabulary words that you understand.
similar polygons
radius
perimeter
area
Rate how well you can find the perimeters and areas of similar polygons.
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265
Lesson 10-4
10-5
Trigonometry and Area
Vocabulary
Review
1. Underline the correct word to complete the sentence.
Area is the number of cubic / square units needed to cover a given surface.
2. Circle the formula for the area of a triangle.
A 5 bh
1
1
A 5 2 bh
1
A 5 2 h(b1 1 b2)
A 5 2 d1d2
Vocabulary Builder
trigonometry (noun) trig uh NAHM uh tree
Other Word Form: trigonometric (adjective)
Definition: Trigonometry is the study of the relationships among two sides and an
angle in a right triangle.
Main Idea: You can use trigonometry to find the area of a regular polygon.
Use Your Vocabulary
Complete each sentence with the word trigonometry or trigonometric.
3. The sine, cosine, and tangent ratios are 9 ratios.
trigonometric
4. This year I am studying 9 in math.
trigonometry
Draw a line from each trigonometric ratio in Column A to its name in Column B.
Column A
length of opposite leg
5. length of hypotenuse
length of adjacent leg
6. length of hypotenuse
length of opposite leg
7. length of adjacent leg
Chapter 10
Column B
cosine
sine
tangent
266
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Related Words: cosine, sine, tangent
Problem 1 Finding Area
Got It? What is the area of a regular pentagon with 4-in. sides? Round your answer
to the nearest square inch.
8. Underline the correct words to complete the sentence.
To find the area using the formula A 5 12 ap, you need to know the length of the
apothem / radius and the perimeter / width of the pentagon.
9. In the regular pentagon at the right, label center C, apothem CR,
and radii CD and CE.
10. The perimeter of the pentagon is 5 ?
11. The measure of central angle DCE is
4
4 in.
C
in., or 20 in.
360
, or 72 .
5
D
Complete Exercises 12 and 13.
12. m/DCR 5 12 m/DCE
5 12 ? 72
R
E
13. DR 5 12 DE
5 12 ? 4
5 36
5
2
14. Use your results from Exercises 12 and 13 to label the diagram below.
C
a
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
D
R
36î
2 in.
15. Circle the equation you can use to find the apothem a.
36
2a
tan 728 5 a
tan 368 5 a
tan 368 5 a2
a
tan 368 5 2
tan 728 5 a2
16. Use the justifications below to find the apothem and the area.
tan 728 5
a ? tan 368 5
2
Use the tangent ratio.
a
Multiply each side by a.
2
2
a 5 tan 368
A 5 12 ap
5 12 ?
<
Divide each side by tan 368.
Write the formula for the area of a regular polygon.
2
tan 368
? 20
27.52763841
Substitute for a and p.
Use a calculator.
17. To the nearest square inch, the area of the regular pentagon is 28 in.2 .
267
Lesson 10-5
Problem 2
Finding Area
Got It? A tabletop has the shape of a regular decagon with a radius of 9.5 in.
What is the area of the tabletop to the nearest square inch?
18. Complete the problem-solving model below.
Know
The radius and
number of sides of the
decagon
Plan
Use trigonometric ratios
to find the apothem and
the length of a side.
Need
The apothem and the
length of a side
19. Look at the decagon at the right. Explain why the measure of each
central angle of a decagon is 36 and m/C is 18.
There are 10 central angles and 360
10 5 36. The apothem
___________________________________________________________________
C
a bisects a central angle, so mlC 5 12(36), or 18.
___________________________________________________________________
20. Use the cosine ratio to find the apothem a.
a
cos 188 5
9.5
9.5
9.5
x
21. Use the sine ratio to find x.
sin 188 5
a
x
9.5
? cos 188 5 a
9.5
? sin 188 5 x
22. Use the justifications below to find the perimeter.
p 5 10 ? length of one side
2
?x
5 10 ?
2
?
The length of each side is 2x.
9.5(sin 188)
5 190 ? sin 188
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
5 10 ?
perimeter 5 number of sides times length of one side
Substitute for x.
Simplify.
23. Find the area. Use a calculator.
A 5 12 ap
5 12 ? 9.5(cos 188) ? 190(sin 188) N 265.2380947
24. To the nearest square inch, the area of the tabletop is 265 in.2 .
Theorem 10-8 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 n ABC 5 12 bc(sin A )
Chapter 10
a
c
25. Complete the formula below.
A
268
b
C
Problem 3 Finding Area
Got It? What is the area of the triangle? Round your answer
10 in.
to the nearest square inch.
34
26. Complete the reasoning model below.
16 in.
Write
Think
I know the lengths of two
Side lengths: 10 in. and 16 in.
sides and the measure of the
included angle.
Angle measure: 34
I can use the formula for the area
Aâ
of a triangle given SAS.
1 Ƃ
10 Ƃ 16 Ƃ sin 34
2
Ƽ 44.73543228
27. To the nearest square inch, the area of the triangle is 45 in.2 .
Lesson Check • Do you UNDERSTAND?
Error Analysis Your classmate needs to find the area of a regular pentagon
with 8-cm sides. To find the apothem, he sets up and solves a trigonometric
ratio. What error did he make? Explain.
a
tan 36
4
a 4 tan 36
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28. The lengths of the legs of the triangle in the regular pentagon
are a and 4 cm.
length of opposite leg
29. The tangent of the 368 angle is length of adjacent leg , or
4
36í
.
a
a
8 cm
30. Explain the error your classmate made.
He used a4 instead of 4a for the tangent ratio.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
area
trigonometry
Rate how well you can use trigonometry to find area.
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269
Lesson 10-5
10-6
Circles and Arcs
Vocabulary
Review
1. Is a circle a two-dimensional figure?
Yes / No
2. Is a circle a polygon?
Yes / No
3. Is every point on a circle the same distance from the center?
Yes / No
4. Circle the figure that is a circle.
Vocabulary Builder
Definition: An arc is part of a circle.
B
Related Words: minor arc, major arc, semicircle
O
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Major arc ABC
Minor arc AC
arc (noun) ahrk
A
C
Example: Semicircle AB is an arc of the circle.
Use Your Vocabulary
Underline the correct word to complete each sentence.
5. A minor arc is larger / smaller than a semicircle.
6. A major arc is larger / smaller than a semicircle.
7. You use two / three points to name a major arc.
8. You use two / three points to name a minor arc.
9. Circle the name of the red arc.
0
0
JK
KL
10. Circle the name of the blue arc.
0
0
KL
JK
Chapter 10
1
LJK
1
LKJ
J
O
1
LJK
270
1
LKJ
L
K
Problem 1 Naming Arcs
Got It? What are the minor arcs of (A?
Draw a line from each central angle in Column A to its
corresponding minor arc in Column B.
Column A
11. /PAQ
12. /QAR
13. /RAS
14. /SAP
15. /SAQ
16. The minor arcs of (A are
Column B
0
RS
0
SP
0
PQ
0
QR
0
SQ
0
0
PQ , QR
P
S
Q
A
R
0
RS
,
,
0
SP
, and
0
SQ
.
Key Concepts Arc Measure and Postulate 10-2
Arc Measure
The measure of a minor arc is equal to the measure of its corresponding central angle.
The measure of a major arc is the measure of the related minor arc subtracted from 360.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
The measure of a semicircle is 180.
Use (S at the right for Exercises 17 and 18.
0
17. m RT 5 m/RST 5 50
1
0
18. m TQR 5 360 2 m RT 5 360 2 50 5 310
R
S
50
T
Q
Postulate 10-2 Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.
1
0
0
m ABC 5 m AB 1 m BC
C
B
Use the circle at the right for Exercises 19 and 20.
0
0
1
A
19. If m AB 5 40 and m BC 5 100, then m ABC 5 140 .
0
0
1
20. If m AB 5 x and m BC 5 y, then m ABC 5
x1y .
Problem 2 Finding the Measures of Arcs
001
1
Got It? What are the measures of PR , RS , PRQ , and PQR in (C?
R
P
77
C
Complete.
0
21. m/PCR 5 77 , so m PR 5 77 .
S
Q 28
271
Lesson 10-6
22. m/RCS 5 m/PCS 2 m/PCR
R
P
77
5 180 2 77 5 103
C
0
23. m/RCS 5 103 , so m RS 5 103 .
1
0
0
0
24. mPRQ 5 m PR 1 m RS 1 m SQ
S
Q 28
5 77 1 103 1 28 5 208
1
0
25. mPQR 5 360 2 m PR
5 360 2 77 5 283
Theorem 10-9 Circumference of a Circle
The circumference of a circle is p times the diameter.
d O
C 5 pd or C 5 2pr
26. Explain why you can use either C 5 pd or C 5 2pr to find the
circumference of a circle.
r
C
Explanations may vary. Sample: The length of the
_______________________________________________________________________
diameter is twice the length of the radius.
_______________________________________________________________________
Got It? A car has a circular turning radius of 16.1 ft. The distance
between the two front tires is 4.7 ft. How much farther does a
tire on the outside of the turn travel than a tire on the inside?
27. The two circles have the same center. To find
the radius of the inner circle, do you
add or subtract?
16.1 ft
subtract
Complete.
28. radius of outer circle 5
16.1
radius of inner circle 5
16.1
4.7 ft
2 4.7 5
11.4
29. circumference of outer circle 5 2pr 5 2p ?
16.1
5
32.2
? p
circumference of inner circle 5 2pr 5 2p ?
11.4
5
22.8
? p
30. Find the differences in the two distances traveled. Use a calculator.
32.2
? p 2
22.8
? p 5
<
9.4
? p
29.53097094
31. A tire on the outer circle travels about 30 ft farther.
Chapter 10
272
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Problem 3 Finding a Distance
Theorem 10-10 Arc Length
The length of an arc of a circle is the product of the ratio
the circumference of the circle.
measure of the arc
and
360
32. Complete the formula below.
0
0
0
m AB
m AB
length of AB
5 360 ? 2πr 5 360 ? πd
Write T for true or F for false.
T
33. The length of an arc is a fraction of the circumference of a circle.
T
0
34. In (O, m AB 5 m/AOB.
Lesson Check • Do you UNDERSTAND?
0
Error Analysis Your class must find the length of AB . A classmate submits
the following solution. What is the error?
B
A
mAB
· 2)r
360
= 110 · 2)(4)
360
= 22 ) m
9
O
C
4m
Length of AB =
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
70
0
35. Is AC a semicircle?
0
36. Does m AB 5 180 2 70 5 110?
Yes / No
Yes / No
37. Is the length of the radius 4?
Yes / No
38. What is the error? Answers may vary. Sample:
The student substituted the diameter instead of the radius for r.
__________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
circle
minor arc
major arc
circumference
Rate how well you can use central angles, arcs, and circumference.
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273
Lesson 10-6
Areas of Circles and
Sectors
10-7
Vocabulary
Review
1. Explain how the area of a figure is different from the perimeter of the figure.
Area is the measure of the space inside a figure while
_________________________________________________________________________
perimeter is the distance around the figure.
_________________________________________________________________________
2. Circle the formula for the area of a parallelogram.
A 5 12 bh
A 5 bh
A 5 12 h(b1 1 b2)
A 5 12 d1d2
3. Find the area of each figure.
6 ft
6 cm
5m
9 cm
A 5 15 m 2
10 ft
A 5 27 cm 2
A 5 60 ft 2
Vocabulary Builder
sector (noun)
SEK
sector RST
R
T
tur
Definition: A sector of a circle is a region bounded by an arc of the
circle and the two radii to the arc’s endpoints.
S
Main Idea: The area of a sector is a fractional part of the area of a circle..
A
Use Your Vocabulary
4. Name the arc and the radii that are the boundaries of the shaded sector.
arc AB
radii CA and CB
C
5. Circle the name of the shaded sector.
sector ABC
sector ACB
sector BAC
6. The shaded sector is what fractional part of the area of the circle? Explain.
90
1 0
1
4 ; AB is a 908 arc and 360 5 4 .
_________________________________________________________________________
Chapter 10
274
B
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
3m
Theorem 10-11 Area of a Circle
The area of a circle is the product of p and the square of the radius.
A 5 pr 2
r
O
Complete each statement.
7. If the radius is 5 ft, then A 5 p ? 5
? 5 .
8. If the diameter is 1.8 cm, then A 5 p ? 0.9 ? 0.9 .
Problem 1 Finding the Area of a Circle
Got It? What is the area of a circular wrestling region with a
42-ft diameter?
9. The radius of the wrestling region is 21 ft.
42 ft
10. Complete the reasoning model below.
Write
Think
I can use the formula for the area
A âπr 2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
of a circle.
I can subtitute the radius into
âπƂ 21
the formula and then simplify.
â 441 Ƃ π
I can use a calculator to find the
approximate area.
2
Ƽ 1385.44236
11. The area of the wrestling region is about
1385
ft2 .
Theorem 10-12 Area of a Sector of a Circle
The area of a sector of a circle is the product of the ratio
and the area of the circle.
measure of the arc
360
A
r
0
m AB
Area of sector AOB 5 360 ? pr2
O
r
B
Complete.
measure of the arc
12. 60
measure of the arc
360
1
60
360 5
6
120
13. 120
360
5
area of the sector
1
6
1
1
3
3
275
?
π
? r2
?
π
? r2
Lesson 10-7
Problem 2
Finding the Area of a Sector of a Circle
Got It? A circle has a radius of 4 in. What is the area of a sector bounded by
a 458 minor arc? Leave your answer in terms of π.
14. At the right is one student’s solution.
What error did the student make?
The student did not square the radius.
______________________________________________________
______________________________________________________
area of sector = 45 ●π(4)
360
1
= ●π(4)
8
1
= π
2
15. Find the area of the sector correctly.
45
area of sector 5 360
? π(4)2
5 18 ? π(16)
5 2π
16. The area of the sector is 2π in.2 .
Key Concept Area of a Segment
Area of sector
Problem 3
Area of triangle
Area of segment
Finding the Area of a Segment of a Circle
Got It? What is the area of the shaded segment shown at the right? Round your
answer to the nearest tenth.
90
5 360 ? p( 4 )2
Substitute.
5
Simplify.
? p
18. nPQR is a right triangle, so the base is 4 m and the height is 4 m.
Chapter 10
P
Q
17. Use the justifications below to find the area of sector PQR.
0
m PR
area of sector PQR 5
? pr2
Use the formula for the area of a sector.
360
4
4m
276
R
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
The area of a segment is the difference of the area of the sector and the area of the
triangle formed by the radii and the segment joining the endpoints.
19. Find the area of nPQR.
A 5 12 bh
5 12 (4)(4)
5 8
20. Complete to find the area of the shaded segment. Use a calculator.
area of shaded segment 5 area of sector PQR 2 area of nPQR
5
4
?p 2
8
< 4.566370614
21. To the nearest tenth, the area of the shaded segment is 4.6 m 2 .
Lesson Check • Do you UNDERSTAND?
Reasoning Suppose a sector in (P has the same area as a sector in (O. Can you
conclude that (P and (O have the same area? Explain.
C
Use the figures at the right for Exercises 22–24.
22. Find the area of sector AOC in (O.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
area of sector 5
45
360
?
23. Find the area of sector RPT in (P.
π(8)2
area of sector 5
180
360
O
45í
8m
A
? π(4)2
5 18 ? π(64)
5 12 ? π(16)
5 8π
5 8π
T
8m
P
R
24. Do the sectors have the same area? Can you conclude that the circles have the
same area? Explain. Explanations may vary. Sample:
Yes; no; the circles do not have the same area because
_______________________________________________________________________
their radii are different lengths.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
sector of a circle
segment of a circle
area of a circle
Rate how well you can find areas of circles, sectors, and segments.
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Lesson 10-7
10-8
Geometric Probability
Vocabulary
Review
Write T for true or F for false.
T
1. A point indicates a location and has no size.
F
2. A line contains a finite number of points.
3. Use the diagram at the right. Circle the segment that includes point S.
PR
PT
P
Q
R
S
T
QR
Vocabulary Builder
Related Term: geometric probability
theoretical probability
P(event) â
number of favorable outcomes
number of possible outcomes
Definition: The probability of an event is the likelihood
keli
ke
l hood that the event will occur
occur.
Main Idea: In geometric probability, favorable outcomes and possible outcomes
are geometric measures such as lengths of segments or areas of regions.
Use Your Vocabulary
4. Underline the correct words to complete the sentence.
The probability of an event is the ratio of the number of favorable / possible
outcomes to the number of favorable / possible outcomes.
5. There are 7 red marbles and 3 green marbles in a bag. One marble is chosen at
random. Write the probability that a green marble is chosen.
P(green)
Write as a fraction.
Write as a decimal.
Write as a percent.
3
10
0.3
30 %
Chapter 10
278
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
probability (noun) prah buh BIL uh tee
Key Concept Probability and Length or Area
Probability and Length
Point S on AD is chosen at random. The probability that S is on BC is the
ratio of the length of BC to the length of AD.
B
A
C
D
BC
P(S on BC) 5 AD
Complete.
AC
6. P(S on AC ) 5
AD
AB
7. P(S on AB) 5
AD
Probability and Area
Point S in region R is chosen at random. The probability that S is in
region N is the ratio of the area of region N to the area of region R.
R
N
area of region N
P(S in region N) 5 area of region R
8. Find the probability for the given areas.
area of region R 5 11 cm2
P(S in N) 5
area of region N 5 3 cm2
3
11
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Problem 1 Using Segments to Find Probability
Got It? Point H on ST is selected at random.
S
What is the probability that H lies on SR?
2
Q
3
4
R
5
6
7
2
2 14
T
8
9
10 11 12 13 14
9. Find the length of each segment.
u
length of SR 5 2 2
8
u
5
length of ST 5
6
u
u
5 12
10. Find the probability.
P(H on SR) 5
length of SR
length of ST
6
5
12
5
1
2
1
11. The probability that H is on SR is 2 , or 50 %.
Problem 2 Using Segments to Find Probability
Got It? Transportation A commuter train runs every 25 min. If a commuter arrives
at the station at a random time, what is the probability that the commuter will have
to wait no more than 5 min for the train?
12. Circle the time t (in minutes) before the train arrives that the commuter will need to
arrive in order to wait no more than 5 minutes.
0#t#5
5 , t # 10
10 , t # 15
15 , t # 20
20 , t # 25
279
Lesson 10-8
13. Circle the diagram that models the situation.
A
C
B
A
C
B
A
0
5
10 15 20 25
0
5
10 15 20 25
0
C
5
B
10 15 20 25
14. Complete.
length of favorable segment 5
length of entire segment 5 25
5
15. Find the probability.
P(waiting no more than 5 min) 5
length of favorable segment
length of entire segment
5
5
1
, or
25
5
1
16. The probability of waiting no more than 5 min for the train is 5 , or 20 %.
Problem 3 Using Area to Find Probability
Got It? A triangle is inscribed in a square.
Point T in the square is selected at random.
What is the probability that T lies in the
shaded region?
5 in.
17. Complete the model below to write an equation.
Let s âthe area of the shaded region.
Relate
area of the
shaded region
is
Write
s
â
area of the
square
area of the
triangle
minus
2
1
Ƃ 5
2
Ľ
5
Ƃ
5
18. Now solve the equation to find the area of the shaded region.
s 5 52 2 12 ? 5 ? 5
5 25 2 12 ? 25
5 25 2 12.5
5 12.5
19. Find the probability.
P(point T is in shaded region) 5
5
area of shaded region
area of square
12.5
1
, or
25
2
1
20. The probability that T lies in the shaded region is 2 , or 50 %.
Chapter 10
280
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Define
Problem 4 Using Area to Find Probability
Got It? Archery An archery target has 5 colored scoring zones formed by
concentric circles. The target’s diameter is 122 cm. The radius of the yellow zone
is 12.2 cm. The width of each of the other zones is also 12.2 cm. If an arrow hits the
target at a random point, what is the probability that it hits the yellow zone?
122
, or 61 cm.
2
22. Find the probability. Write the probability as a decimal.
21. The radius of the target is
area of yellow zone
P(arrow hits yellow zone) 5 area of entire target
5
p(12.2)2
p( 61 )2
148.84
5
5
0.04
3721
23. Explain why the calculation with p is not an estimate. Answers may vary. Sample:
I divide π out before finding the quotient.
_______________________________________________________________________
24. The probability that the arrow hits the yellow zone is
0.04 , or 4 %.
Lesson Check • Do you UNDERSTAND?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
SQ
Reasoning In the figure at the right, QT 5 12 . What is the probability that
a point on ST chosen at random will lie on QT ? Explain.
S
Q
T
25. If SQ 5 x, then QT 5 2x and ST 5 3x .
26. What is P(point on QT )? Explain.
2
3;
length of QT
2
5 2x
3x 5 3
length
of
ST
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
length
area
geometric probability
Rate how well you can use geometric probability.
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Lesson 10-8