Lesson 12.1 Assignment

Lesson 12.1  Assignment Name_________________________________________________________ Date__________________________ Introduction to Circles Circle, ...
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Lesson 12.1  Assignment Name_________________________________________________________ Date__________________________

Introduction to Circles Circle, Radius, and Diameter Miguel is the defending watermelon seed spitting champion. He can consistently spit a watermelon seed 6 feet. 1. Miguel practices in his backyard to make sure he can defeat his competition again this year. a. Miguel stands in the middle of his backyard. Draw a point in the middle of the space and label the point M.

A B

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M

Chapter 12      Assignments      •      221

Lesson 12.1  Assignment

page 2

b. Miguel spits a watermelon seed. Let 6 centimeters represent the 6 feet that the watermelon seed travels. Use a centimeter ruler to locate and draw a point that is 6 centimeters from point M. Label this point A.

Answers will vary. See picture.

c. Standing in the same spot, Miguel spits a second watermelon seed in a different direction. Locate another point that is exactly 6 centimeters from point M. Label this point B.

Answers will vary. See picture.

d. Miguel continues to practice. Continue to draw points exactly 6 centimeters from point M until you have at least 10 distinct points.

Answers will vary. See picture.

e. How would you describe the shape that is formed by connecting all of the points representing seeds located exactly 6 centimeters from where Miguel is standing?

A circle is formed by connecting all of the points representing seeds located exactly 6 centimeters from where Miguel is standing.

f. What is the radius of the circle formed by the seeds Miguel is spitting in his backyard? Explain your reasoning.

The radius of the circle is 6 feet. The radius of the circle is the distance from the center of the circle where Miguel is standing to a point on the circle where the watermelon seeds land.

g. What is the diameter of the circle formed by the seeds Miguel is spitting in the backyard? Explain your reasoning.

The diameter is 12 feet. The diameter is twice the length of the radius.

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This distance is 6 feet.

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Lesson 12.1  Assignment

page 3

Name_________________________________________________________ Date__________________________ 2. Miguel and his brother, Tomas, can each spit a watermelon seed 6 feet. He joins Miguel in the backyard to help him practice for a competition. Miguel and Tomas are standing 6 feet apart. The picture shows a model of the circles formed by connecting the seeds that each boy spits.

© 2011 Carnegie Learning

M

T

a. What can you conclude about Circle M and Circle T? Explain your reasoning.

Circle M and Circle T are congruent circles. Radius MT is the same radius for both circles.

b. What can you conclude about the lengths of the diameters of Circle M and Circle T? Explain your reasoning.

The lengths of the diameters are 12 feet for both circles. The diameters of congruent circles are the same length.

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Lesson 12.1  Assignment

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c. Miguel and Tomas, who can both spit watermelon seeds 6 feet, have been practicing their spitting and are now checking out where their seeds landed. They found that 2 of their seeds landed in the exact same spot. Let S represent the point where both seeds landed. Connect the points that represent Miguel, Tomas, and the spot where the seeds landed.

M

T

d. What can you conclude about the shape that is formed by Miguel, Tomas, and the spot where the seeds landed? Explain your reasoning.

Triangle MTS is an equilateral triangle. Line segments MT, MS, and TS are the same length because they are all radii of two congruent circles.

© 2011 Carnegie Learning

S

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Lesson 12.2  Assignment Name_________________________________________________________ Date__________________________

But Most of All, I Like Pi! Circumference of a Circle 1. Although she’s only in middle school, Tameka loves to drive go-carts! Her favorite place to drive go-carts, Driver’s Delight, has 3 circular tracks. Track #1 has a radius of 60 feet. Track #2 has a radius of 85 feet. Track #3 has a radius of 110 feet. a. Compute the circumference of Track #1 using the circumference formula. Let π 5 3.14.

C 5 2πr



C 5 2(3.14)(60)



C 5 376.8



The circumference of Track #1 is 376.8 feet.

© 2011 Carnegie Learning

b. Compute the circumference of Track #2 using the circumference formula. Let π 5 3.14.

C 5 2πr



C 5 2(3.14)(85)



C 5 533.8



The circumference of Track #2 is 533.8 feet.

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Lesson 12.2  Assignment

page 2

c. Compute the circumference of Track #3 using the circumference formula. Let π 5 3.14.

C 5 2πr



C 5 2(3.14)(110)



C 5 690.8



The circumference of Track #3 is 690.8 feet.

d. Driver’s Delight is considering building a new track. They have a circular space with a diameter of 150 feet. Compute the circumference of the circular space. Let π 5 3.14. C 5 πd



C 5 (3.14)(150)



C 5 471



The circumference of the space is 471 feet.

© 2011 Carnegie Learning



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Lesson 12.2  Assignment

page 3

Name_________________________________________________________ Date__________________________ 5. Tameka wants to build a circular go-cart track in her backyard. a. If she wants the track to have a circumference of 150 feet, what does the radius of the track need to be? Round your answer to the nearest hundredth, if necessary.

C 5 2πr



150 5 2(3.14)r



150 5 6.28r



6.28r 150    _____ ​   ​ 5 ​  ​ _____ 6.28 6.28



23.89 ¯ r



The radius of the track should be approximately 23.89 feet.

b. If she wants the track to have a circumference of 200 feet, what does the radius of the track need to be? Round your answer to the nearest hundredth, if necessary.

© 2011 Carnegie Learning



C 5 2πr



200 5 2(3.14)r



200 5 6.28r



6.28r 200    _____ ​   ​ 5 ​  ​ _____ 6.28 6.28



31.85 ¯ r



The radius of the track should be approximately 31.85 feet.

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Lesson 12.2  Assignment

page 4

c. If she wants the track to have a circumference of 400 feet, what does the diameter of the track need to be? Round your answer to the nearest hundredth, if necessary.

C 5 πd



400 5 3.14d



3.14d 400    ______ ​   ​ 5 ​  ​ _____ 3.14 3.14



127.39 ¯ d

© 2011 Carnegie Learning

The diameter of the track should be approximately 127.39 feet.

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Lesson 12.3  Assignment Name_________________________________________________________ Date__________________________

One Million Sides Area of a Circle 1. Inscribed Circle D intersects the regular octagon at the midpoint of each side. The radius of the circle is r, and the length of each side of the octagon is s, as shown.

D

r s

a. Draw 8 line segments from the center point of the circle to each vertex of the octagon to form 8 congruent triangles. How is the radius of the circle, r, related to the 8 triangles?

The radius of the circle is also the height of each triangle.

© 2011 Carnegie Learning

b. Write a formula to describe the area of each of the 8 triangles.

1  ​bh A 5 ​ __ 2



1 ​sr A 5 ​ __ 2

c. Write a formula to describe the area of the octagon.

(  )



A 5 8 ​ __ ​ 1 ​sr  ​ 2



A 5 4sr

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Lesson 12.3  Assignment

page 2

d. Write a formula to describe the perimeter of the octagon.

P 5 8s

e. Write a formula to describe the area of the pentagon in terms of the perimeter.

(  )

1 ​ Pr. P ​ . ​ P ​   ​r or A 5 ​ __   If P 5 8s, then s 5 ​ __   So, A 5 4​ __ 8 8 2

© 2011 Carnegie Learning



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Lesson 12.4  Assignment Name_________________________________________________________ Date__________________________

It’s About Circles! Unknown Measurements Jamal loves his dog, Rupert. On sunny days, Jamal keeps Rupert on a 12-foot leash in the backyard. The leash is secured to a stake in the ground. 1. Draw a picture to represent all of the area where Rupert can play. Label the radius of the circle.

12 ft

a. What is the diameter of Rupert’s play area? Explain your reasoning.

The diameter of Rupert’s play area is 24 feet. The diameter is double the radius.

© 2011 Carnegie Learning

b. What is the circumference of Rupert’s play area? Use 3.14 for π.

C 5 2πr



C 5 2(3.14)(12)



C 5 75.36



The circumference of Rupert’s play area is 75.36 feet.

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Lesson 12.4  Assignment

page 2

c. What is the area of Rupert’s play area? Use 3.14 for π.

A 5 πr2



A 5 (3.14)(12)2



A 5 452.16



The area of Rupert’s play area is 452.16 square feet.

d. Suppose Jamal wants to give Rupert a little more room to play. He uses a 15-foot leash instead of the usual leash. What is the area of Rupert’s play area now? Use 3.14 for π. A 5 πr2



A 5 (3.14)(15)2



A 5 706.5



The area of Rupert’s play area is 706.5 square feet.

© 2011 Carnegie Learning



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Lesson 12.4  Assignment

page 3

Name_________________________________________________________ Date__________________________ 2. Jamal would like to build a square fence around his dog Rupert’s play area without infringing on the dog’s space.

12 ft

a. What is the area of the fenced-in space? A 5 s2



A 5 242



A 5 576



The area of the fenced-in space is 576 square feet.

© 2011 Carnegie Learning



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Lesson 12.4  Assignment

page 4

b. What is the area of the region in the fenced-in space that Rupert cannot reach when he’s on the leash?

Area of the square: 576 ft2



A 5 πr2



A 5 (3.14)(12)2



A 5 452.16



Area of the circle: 452.16 ft2



Area of the shaded region 5 Area of the square 2 Area of the circle



5 576 2 452.16



5 123.84 The area of the shaded region is 123.84 square feet.

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Lesson 12.4  Assignment

page 5

Name_________________________________________________________ Date__________________________ 3. Jamal is dog-sitting his friend’s dog, Rufus, who has a 10-foot leash. Because Jamal’s dog Rupert and Rufus tend to fight when they hang out together, Jamal places their leashes in the backyard so that they are as close as possible without overlapping. Determine the area of the shaded region.

12 ft

10 ft

Area of Rupert’s play area: A 5 πr2

A 5 (3.14)(12)2



A 5 452.16

© 2011 Carnegie Learning

Area of Rufus’ play area: A 5 πr2

A 5 (3.14)(10)2



A 5 314

Area of the rectangle: A 5 (24 1 20)(24)

A 5 1056

Area of the shaded region 5 Area of the rectangle 2 (Rupert’s play area 1 Rufus’ play area)

5 1056 2 (452.16 1 314)



5 289.84

The area of the shaded region is 289.84 square feet. Chapter 12      Assignments      •      235

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236      •      Chapter 12      Assignments

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