student workbook with scaffolded practice unit 6 · 2017. 8. 9. · given that all circles are...

Student Workbookwith Scaffolded Practice

Unit 6

1

1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-7771-2

Copyright © 2014

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

EDUCATIONWALCH

This book is licensed for a single student’s use only. The reproduction of any part, for any purpose, is strictly prohibited.

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Officers. All rights reserved.

2

Program pages

Workbook pages

Introduction 5

Unit 6: Circles With and Without CoordinatesLesson 1: Introducing Circles

Lesson 6.1.1: Similar Circles and Central and Inscribed Angles . . . . . . . . . . . . . . .U6-5–U6-27 7–18

Lesson 6.1.2: Chord Central Angles Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . .U6-28–U6-42 19–30

Lesson 6.1.3: Properties of Tangents of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . .U6-43–U6-62 31–44

Lesson 2: Inscribed Polygons and Circumscribed TrianglesLesson 6.2.1: Constructing Inscribed Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U6-71–U6-95 45–56

Lesson 6.2.2: Constructing Circumscribed Circles . . . . . . . . . . . . . . . . . . . . . . . .U6-96–U6-112 57–68

Lesson 6.2.3: Proving Properties of Inscribed Quadrilaterals . . . . . . . . . . . . . U6-113–U6-127 69–80

Lesson 3: Constructing Tangent LinesLesson 6.3.1: Constructing Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-136–U6-161 81–92

Lesson 4: Finding Arc Lengths and Areas of SectorsLesson 6.4.1: Defining Radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-169–U6-181 93–102

Lesson 6.4.2: Deriving the Formula for the Area of a Sector . . . . . . . . . . . . . . U6-182–U6-193 103–112

Lesson 5: Explaining and Applying Area and Volume FormulasLesson 6.5.1: Circumference and Area of a Circle . . . . . . . . . . . . . . . . . . . . . . . U6-200–U6-221 113–122

Lesson 6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres . . . . . . . U6-222–U6-244 123–132

Lesson 6: Deriving EquationsLesson 6.6.1: Deriving the Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . U6-252–U6-277 133–142

Lesson 6.6.2: Deriving the Equation of a Parabola . . . . . . . . . . . . . . . . . . . . . . U6-278–U6-305 143–154

Lesson 7: Using Coordinates to Prove Geometric Theorems About Circles and ParabolasLesson 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-313–U6-339 155–166

Station ActivitiesSet 1: Circumference, Angles, Arcs, Chords, and Inscribed Angles. . . . . . . . . U6-367–U6-376 167–176

Set 2: Special Segments, Angle Measurements, and Equations of Circles . . . U6-382–U6-388 177–184

Set 3: Circumcenter, Incenter, Orthocenter, and Centroid . . . . . . . . . . . . . . . U6-395–U6-402 185–192

Coordinate Planes 193–220

Formulas 221–226

Bilingual Glossary 227–268

Table of Contents

CCSS IP Math II Teacher Resource© Walch Educationiii

3

The CCSS Mathematics II Student Workbook with Scaffolded Practice includes all of the student pages from the Teacher Resource necessary for your day-to-day classroom use. This includes:

• Warm-Ups

• Problem-Based Tasks

• Practice Problems

• Station Activity Worksheets

In addition, it provides Scaffolded Guided Practice examples that parallel the examples in the TRB and SRB. This supports:

• Taking notes during class

• Working problems for preview or additional practice

The workbook includes the first Guided Practice example with step-by-step prompts for solving, and the remaining Guided Practice examples without prompts. Sections for you to take notes are provided at the end of each sub-lesson. Additionally, blank coordinate planes are included at the end of the full unit, should you need to graph.

The workbook is printed on perforated paper so you can submit your assignments and three-hole punched to let you store it in a binder.

CCSS IP Math II Teacher Resource© Walch Educationv

Introduction

5

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESLesson 1: Introducing Circles

U6-5© Walch Education CCSS IP Math II Teacher Resource

6.1.1

Name: Date:

Gunner is a ceramics artist. He wants to create two proportional triangular plates. The dimensions of one of the plates are given.

C

7 in

7.5 in5 in

B

A D

F

E

3 in

1. What is the scale factor Gunner used to create the smaller plate?

2. What are the lengths of the two missing sides?

Lesson 6.1.1: Similar Circles and Central and Inscribed Angles

Warm-Up 6.1.1

7

Unit 6 • CirCles with and without CoordinatesLesson 1: Introducing Circles

U6-13CCSS IP Math II Teacher Resource

6.1.1© Walch Education

Name: Date:

Scaffolded Practice 6.1.1Example 1

Prove that the measure of a central angle is twice the measure of an inscribed angle that intercepts the same arc.

Given: A with inscribed ∠B and central ∠CAD intercepting CD .

Prove: 2m B m CAD∠ = ∠

1. Identify the known information.

2. Identify what information is known about the angles of the triangle.

continued

9


U6-14CCSS IP Math II Teacher Resource 6.1.1

© Walch Education

Name: Date:

Example 2

Prove that all circles are similar using the concept of similarity transformations.

Example 3

A car has a circular turning radius of 15.5 feet. The distance between the two front tires is 5.4 feet. To the nearest foot, how much farther does a tire on the outer edge of the turning radius travel than a tire on the inner edge?

15.5 ft

5.4 ft

Example 4

Find the value of each variable.

D F

E

C

a˚

b˚

c˚70˚

104˚

Example 5

Find the measures of ∠BAC and ∠BDC .

C

D

B

A(x + 14)˚

(7x – 7)˚

10


U6-22© Walch EducationCCSS IP Math II Teacher Resource

6.1.1

Name: Date:

Problem-Based Task 6.1.1: Following in Archimedes’ FootstepsThe precise determination of the value of pi was of interest to the ancient Greeks for many years. Several approximations were calculated, but Archimedes was credited with determining a very precise approximation of pi. Identify several circular objects in your classroom and verify the approximation of a circle’s circumference to its diameter.

Identify several circular objects

in your classroom and verify the approximation of a circle’s

circumference to its diameter.

11



6.1.1

Name: Date:

continued

Given that all circles are similar, determine the scale factor necessary to map A B→ .

1. A has a radius of 4 units and B has a radius of 6 units.

2. A has a diameter of 55 units and B has a diameter of 75 units.

Use your knowledge of similar circles to complete problems 3 and 4.

3. A homeowner hired a landscaper to expand her circular garden. If the landscaper uses a scale

factor of 4

3 to expand the garden, what is the difference in the radii of the new and old garden?

4 ft x

4. A child’s train has a circular turning radius of 12 inches. The distance between the two front tires is 3 inches. To the nearest tenth of an inch, how much farther does the tire on the outer edge of the turn travel than a tire on the inner edge?

Practice 6.1.1: Similar Circles and Central and Inscribed Angles

13



6.1.1

Name: Date:

Use your knowledge of angles to complete the problems that follow.

5. Find the values of x and y.

C

D

A

150˚

32˚

B

y˚

x˚

6. Find the value of x and the measure of AB .

DA

B

(3x + 20)˚

C(5x – 16)˚

7. Find the values of x, y, and z.

x˚63˚

z˚

y˚

continued

14



6.1.1

Name: Date:

8. Find m C∠ and m D∠ .

DA

B

(c + 1)˚

C

(3c – 17)˚

9. Find m B∠ and m C∠ .

(8x – 1)˚

C

D

B

(4x + 7)˚

10. Find mBC and mCA .

48˚

C

A

B

110˚

15

Notes

Name: Date:

17

Notes

Name: Date:

18



6.1.2

Name: Date:

Your classmate Jana has been sick, and is making up several days’ worth of missed homework. Your teacher has asked you to explain the homework to Jana. Use the following diagram to help her.

C

B

A

D

x˚

(x + 12)˚

1. Explain to Jana the relationship between m D∠ and m B∠ .

2. Set up an equation and show Jana how to solve for x.

3. Help Jana find m D∠ and m B∠ .

Lesson 6.1.2: Chord Central Angles Conjecture

Warm-Up 6.1.2

19



© Walch Education

Name: Date:


In A , m BAC∠ = 57 . What is mBDC ?

B

AC

D

57˚

1. Find the measure of BC .

2. Find the measure of BDC .

3. State your conclusion.

continued

21




Name: Date:

Example 2

G E≅ . What conclusions can you make?

K

E J

H

G

I

Example 3

Find the value of y.

CD

AE

B

(8y – 47)˚(13y – 48)˚

22



6.1.2

Name: Date:

In art class, you are creating a mask from a circular plate. The circumference of the plate is 30 centimeters and you have been instructed to cut out the central angle such that its intercepted major arc measures 200º. What is the length of the arc on the angle that you need to cut out?

Problem-Based Task 6.1.2: Masking the Problem

What is the length of the arc on the angle that you need

to cut out?

23



6.1.2

Name: Date:

continued

Use what you’ve learned about chords and central angles to solve.

1. In A , m BAC∠ = 64 . What is mBDC ?

B

CD

A64˚

2. In A , BDC = 238 . What is m BAC∠ ?

B

CD

A238˚

3. What is the value of t?

B

CD

A(2t – 3)˚

(9 – 4t)˚

Practice 6.1.2: Chord Central Angles Conjecture

25



6.1.2

Name: Date:

4. If circles G and E are congruent, what can you conclude about G and E ?

G

I

H

K

JE

5. Is there enough information to conclude that the central angles of the chords are congruent? Explain.

G

I

H

K

JE

6. G E≅ . What is the value of y?

G

I

H

(2y – 6)˚

64˚

K

JE

continued

26



6.1.2

Name: Date:

7. Find the value of t.

A

B

CD

E

(6t – 12)˚ (18 – 4t)˚


A

B

CD

E

(7t + 2)˚

(94 + 3t)˚


A

B

CD

E

(15 – t)˚

(23 – 2t)˚

10. The circumference of the trunk of a tree to be decorated is 12 inches. You have 7 inches of garland to wrap partially around the tree trunk. What is the arc angle of the trunk that you will decorate?

27

Notes

Name: Date:

29

Notes

Name: Date:

30



6.1.3

Name: Date:

Lennon is an artist. He is in the process of designing a logo for the Star model rocket company and has some of the dimensions of his design. Help Lennon make sure his design is accurate by answering the following questions.

y

x

WF

T

R

AS

EL

1. In order to fit in the allotted space, the length of SL must be 24 centimeters and the length of radius RL must be 7 centimeters. If both of the bottom triangles are congruent, and ∠REA is equal to 90º, what must the length of SR be?

2. Based on the information discovered in your answer to the previous question, what can Lennon conclude about a line that touches a circle at exactly one point?

3. Lennon needs to figure out the lengths of TW and TF . The points on the design grid are

T (0, 9), F (–4, 2), and W (4, 2). Use the distance formula, d x x y y= −( ) + −( )2 1 2 2 1 2 , to find the lengths of TW and TF .

4. Based on the information discovered in your answer to the previous problem, what can Lennon conclude the next time he draws two lines that extend from the same point and touch the same circle, each in exactly one point?

Lesson 6.1.3: Properties of Tangents of a Circle

Warm-Up 6.1.3

31




Name: Date:


Determine whether BC is tangent to A in the diagram below.

A B9

C

41 40

1. Identify the radius.

2. Determine the relationship between AB and BC at point B in order for BC to be tangent to A .

3. Show that ∠ABC is a right angle by using the converse of the Pythagorean Theorem.


continued

33



© Walch Education

Name: Date:

Example 2

Each side of ABC is tangent to circle O at the points D, E, and F. Find the perimeter of ABC .

O

EB

D

C

A

7

5

16

F

continued

34




Name: Date:

Example 3

A landscaper wants to build a walkway tangent to the circular park shown in the diagram below. The

other walkway pictured is a radius of the circle and has a slope of −1

2 on the grid. If the walkways

should intersect at (4, –2) on the grid, what equation can the landscaper use to graph the new

walkway on the grid?y

x-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

continued

35



© Walch Education

Name: Date:

Example 4

AB is tangent to C at point B as shown below. Find the length of AB as well as m BD .

C

A

8B

28˚

D

17

36



6.1.3

Name: Date:

Problem-Based Task 6.1.3: The Circus Is in Town! Is It Safe? The circus has just arrived in town and is setting up. The fire inspector is on site to make sure that the circus meets local fire codes. The fire code requires that the circus have at least 175 feet of exterior walkways to pass inspection so that audience members are safe.

In the diagram below, all lines that appear to be tangent are tangent. The distance from the ticket booth, exit A, to the center of the concession stand is 25 feet. It is the same distance from the center of the concession stand to the petting zoo, at exit B. Use this information and the diagram to determine if the circus will pass inspection.

Big top

B

A

13 ft

Petting zoo

Concession stand7 ft

(x + 23) ft

(2x – 7) ft

Ticket booth

Use this information and the diagram to determine if the circus will pass

inspection.

37



6.1.3

Name: Date:

continued

Use what you have learned about tangent lines and secant lines to answer the questions.

1. AB and AC are both tangent to circle D. If AB is 14 units and AC is (x – 3) units, what is the value of x?

2. Is GH tangent to circle F in the diagram below?

F9

H

G

40

41

3. QR is tangent to circle S at point R. What is the length of QR ?

S

37

12

R

Q

Practice 6.1.3: Properties of Tangents of a Circle

39



6.1.3

Name: Date:

4. The class pet is a hamster. It has an exercise wheel like the one pictured below. What is the length of the base of the wheel stand?

8 cm 8 cm5 cm

5. BC is tangent to A at point B in the diagram below. What is the length of BC ?

A

B

20

C

9

6. Emma must prove that AB is tangent to a circle C at point A. She is going to show that ABC is a right triangle. Why is this an important part of her proof?

continued

40



6.1.3

Name: Date:

7. The sides of quadrilateral PQRS are tangent to the circle at the points as pictured below. What is the length of QR ?

R

2

P Q8

3

5

S

8. Pictured below is the logo for a new ice cream shop. The circles are congruent. If the diameter of each circle is 10 feet, will the logo fit on a billboard that is 60 feet tall? Explain.

5x 3x + 18

continued

41



6.1.3

Name: Date:

9. A satellite (S) in orbit around the Earth is sending two signals that are tangent to Earth at points A and B. If m ACB is 208º and m AB is 152º, what is the measure of ∠S ?

C

B

A

S

10. Students in an algebra class are designing a new banner for the school’s mascot, “The Fighting Duck.” The beak is made from two tangents, connected by a vertical line at the point of tangency. If the students only have 200 feet of material for the duck’s beak, what value of x will make the tangent sides the correct length? What is the length of each side?

8x

16 ft

42

Notes

Name: Date:

43

Notes

Name: Date:

44

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESLesson 2: Inscribed Polygons and Circumscribed Triangles


6.2.1

Name: Date:

A hiking trail with a common starting point diverges into two forks in a meadow. You notice uncommon wildflowers that you have wanted to photograph, so you decide to leave the trail.

1. Along what geometric construction would you need to walk in order to stay the same distance from either fork of the trail?

2. The diagrams below use the legs of triangles to show the forks of the trail. Each ray represents a possible path you could take if you leave the trail. Which of the diagrams represents walking along a path that is equidistant from either trail?

A B C

Lesson 6.2.1: Constructing Inscribed Circles

Warm-Up 6.2.1

45

Unit 6 • CirCles with and without CoordinatesLesson 2: Inscribed Polygons and Circumscribed Triangles


© Walch Education

Name: Date:


Verify that the angle bisectors of acute ABC are concurrent and that this concurrent point is equidistant from each side.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

y

x

B (10, 2)

C (–2, –7)

A (–2, 7)

1. Construct the angle bisector of ∠A .

2. Repeat the process for ∠B and ∠C .

3. Locate the point of concurrency. Label this point as D.

4. Verify that the point of concurrency is equidistant from each side.


continued

47




Name: Date:

Example 2

Construct a circle inscribed in acute ABC .

C

A B

Example 3

Construct a circle inscribed in obtuse ABC .

A B

C

48



6.2.1

Name: Date:

Problem-Based Task 6.2.1: First Aid StationOrganizers are setting up a first aid station and roping off paths for a cross-country race. There are three routes that the runners will travel at least twice throughout the race. Where should the organizers place the first aid station so that help can be provided easily? Where should the roped paths be placed for the responders?

East route

South route

North route

Where should the organizers place the

first aid station so that help can be provided easily?

Where should the roped paths

be placed for the responders?

49



6.2.1

Name: Date:

continued

Construct the inscribed circle for each of the triangles in problems 1–3.

1.

2.

3.

4. Must the incenter always be found within the interior of the triangle? Why or why not? Consider your constructions of the previous three problems.

Practice 6.2.1: Constructing Inscribed Circles

51



6.2.1

Name: Date:

Use the diagrams provided to solve problems 5–10.

5. In the map of Georgia below, Interstates 475 and 75 form a triangle with Macon as one of the vertices. A company wants to build its new headquarters in the middle of that triangle so that the building will be equidistant from each interstate. Where should the headquarters be built?

C

B

A

Macon

75

75

475

6. Find the length of BI. Assume that I is the incenter.

4 5

6

E

D

F

I

A

B

C

continued

52



6.2.1

Name: Date:

7. Find the measure of ∠BIC . Assume that BI and CI are angle bisectors.

110°

I

A

C

B

8. Find the measure of ∠AIB . Assume that AI and BI are angle bisectors.

A

B

C

45˚

I

continued

53



6.2.1

Name: Date:

9. Suppose that ABC is isosceles, AB = AC, and point I is the incenter. What is true about BI and CI ? Support your answer.

I

A

C

B

10. What type of quadrilateral is AFDG? Assume that D is the incenter. Support your answer.

E

FG

D

A

B

C

54

Notes

Name: Date:

55

Notes

Name: Date:

56



6.2.2

Name: Date:

Luca is creating an origami boat and has folded a square piece of paper into a triangle. The next step in making the boat requires Luca to fold the triangle such that the fold becomes the perpendicular bisector of the triangle’s longest side.

1. Which figure represents the relationship between the fold and the triangle?

CBA

2. Which of the following is a correct step in constructing a perpendicular bisector of a segment?

a. Make two arcs, centered at each endpoint, that intersect the segment twice.

b. Make two arcs, centered at each endpoint, that intersect each other twice.

c. Use a ruler and a protractor to find the midpoint and draw a right angle.

3. In order to create the fold so that it becomes the perpendicular bisector of the triangle’s longest side, what must be true about the lengths of the paper on either side of the fold?

Lesson 6.2.2: Constructing Circumscribed Circles

Warm-Up 6.2.2

57



© Walch Education

Name: Date:


Verify that the perpendicular bisectors of acute ABC are concurrent and that this concurrent point is equidistant from each vertex.

A

B

C

1. Construct the perpendicular bisector of AB .

2. Repeat the process for BC and AC .

3. Locate the point of concurrency. Label this point D.

4. Verify that the point of concurrency is equidistant from each vertex.

continued

59




Name: Date:

Example 2

Construct a circle circumscribed about acute ABC .

A B

C

Example 3

Construct a circle circumscribed about obtuse ABC .

AB

C

60



6.2.2

Name: Date:

Problem-Based Task 6.2.2: Building a New Radio StationThe owners of a radio station in Georgia want to build a new broadcasting building located within the triangle formed by Atlanta, Columbus, and Macon. Where should the station be built so that it is equidistant from each city?

C

MMacon

Columbus

Atlanta

Woodland

Orchard Hill

Zebulon

A

75

75

75

185

85

85

Where should the station be

built so that it is equidistant from

each city?

61



6.2.2

Name: Date:

continued

Construct the circumscribed circle for each of the triangles in problems 1–3.

1.

2.

3.

Practice 6.2.2: Constructing Circumscribed Circles

63



6.2.2

Name: Date:

Use what you’ve learned and the diagrams, when provided, to complete problems 4–10.

4. The producers of a cooking show on the Snackers Network are designing a new set. A food preparation station needs to be located between the refrigerator, sink, and stove. Which point of concurrency, the circumcenter or the incenter, will result in the preparation station being located in a place that is equidistant from the refrigerator, sink, and stove?

5. Assume that point C is the circumcenter for ABD . What is the length of AB ?

13

12

C

M

NL

A

DB

6. Must the circumcenter be located within the triangle? Why or why not?

7. Offices 1, 4, and 6 are located on the same floor of a renovated building. The offices must be connected by electrical wires, and all the wires will lead to a central control box. In which room should the control box be placed so that the amount of wiring is minimized?

Hallway

Office 1 Office 2 Office 3 Office 4

Office 5 Office 6 Office 7 Office 8

continued

64



6.2.2

Name: Date:

8. Using the diagram below, prove why the circumcenter is equidistant from the vertices of a triangle.

D

O

NM

A B

C

9. Is it possible for the incenter to be the same point as the circumcenter? Why or why not? If it is possible, what type(s) of triangle would meet this criterion? Consider your responses to problems 1 through 3 in determining your answer.

10. Describe a method to determine the center of the circle below. Then, carry out your plan.

65

Notes

Name: Date:

67

Notes

Name: Date:

68



6.2.3

Name: Date:

Jordan is practicing his dunk shot for a basketball contest. He wants to show off by doing a 360º slam. If he can perfect it, he’ll jump, spin in a complete circle, and slam dunk the basketball. However, in practicing, he is only turning to his left 280º.

1. How much farther must Jordan turn in order to achieve his goal of 360º?

2. What fraction of the 360º slam is Jordan currently turning?

Lesson 6.2.3: Proving Properties of Inscribed Quadrilaterals

Warm-Up 6.2.3

69




Name: Date:


Consider the inscribed quadrilateral in the following diagram. What are the relationships between the measures of the angles of an inscribed quadrilateral?

122 °52 °

104 °

82 °

A

B

C

D

1. Find the measure of ∠B .

2. Find the measure of ∠D .

3. What is the relationship between ∠B and ∠D ?

4. Does this same relationship exist between ∠A and ∠C ?


continued

71



© Walch Education

Name: Date:

Example 2

Consider the inscribed quadrilateral below. Do the relationships discovered between the angles in Example 1 still hold for the angles in this quadrilateral?

100 °74 °

104 °

82 °

A

B

C

E

Example 3

Prove that the opposite angles of the given inscribed quadrilateral are supplementary.

z°

y°

x°

w°

B

CD

A

72



6.2.3

Name: Date:

King Arthur is trying to assign seats at his Round Table for three of his knights and himself. If the angle formed by Galahad, Arthur, and Lancelot is 112º, is there enough information to determine the measure of the angle formed by Galahad, Bedivere, and Lancelot? Is there enough information to determine the measures of the angles at Galahad and Lancelot? Also, since Arthur is a geometry fan, he would like to see what types of quadrilaterals he and his knights could form.

112°

LC

G A

B

Problem-Based Task 6.2.3: King Arthur and His Round Table

Is there enough information to determine the

measure of the angle formed by

Galahad, Bedivere, and Lancelot?

73



6.2.3

Name: Date:

continued

Use the provided diagrams and your knowledge of the properties of inscribed quadrilaterals to complete problems 1–5.

1. Find the values of x and y.

°

x°

115 °D

C

A

y

B

2. Find the value of x. Assume that quadrilateral ABCD is a kite.

x°

136 °

B

A

C

D

Practice 6.2.3: Proving Properties of Inscribed Quadrilaterals

75



6.2.3

Name: Date:

3. Is ≅EF FD ? Why or why not?

70 °

70 °

E

C

D

F

4. Assume that quadrilateral ABCD below is a square that is inscribed in the circle. Construct a circle inscribed in the square.

C

A

D

B

continued

76



6.2.3

Name: Date:

5. Find the values of x, y, and z.

z°

°

x°

100 °

y

71 °

68 °

Use the figure below to complete problems 6–8.

51 °

72 °A

D C

B

112 °

E

6. Find the measure of BC .

7. Find the measure of ∠B .

8. Find the measure of ∠BCD .

Use your knowledge of inscribed quadrilaterals to answer questions 9 and 10.

9. Is it possible for a trapezoid to be inscribed in a circle? Justify your answer.

10. If a rectangle is inscribed in a circle, each diagonal also serves another function. What is this function?

77

Notes

Name: Date:

79

Notes

Name: Date:

80

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESLesson 3: Constructing Tangent Lines


6.3.1

Name: Date:

A city is planning to develop a straight trail near a circular playground. Below is a graph of the playground, with the center at (3, –1). The trail is meant to be perpendicular to the playground at the point (6, 3).

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

y

x

1. What must you know about the radius of the circle before you can determine the line that represents the trail? Why?

2. How would you sketch the trail at point (6, 3) so that it is perpendicular to the radius drawn to (6, 3)?

3. What is the equation of the line for the trail?

Lesson 6.3.1: Constructing Tangent Lines

Warm-Up 6.3.1

81

Unit 6 • CirCles with and without CoordinatesLesson 3: Constructing Tangent Lines


© Walch Education

Name: Date:


Use a compass and a straightedge to construct BC tangent to circle A at point B.

A

B

1. Draw a ray from center A through point B and extending beyond point B.

2. Put the sharp point of the compass on point B. Set it to any setting less than the length of AB , and then draw an arc on either side of B, creating points D and E.

3. Put the sharp point of the compass on point D and set it to a width greater than the distance of DB . Make a large arc intersecting

� ��AB .

4. Without changing the compass setting, put the sharp point of the compass on point E and draw a second arc that intersects the first. Label the point of intersection with the arc drawn in step 3 as point C.

5. Draw a line connecting points C and B, creating tangent � ��BC .

continued

83




Name: Date:

Example 2

Using the circle and tangent line from Example 1, construct two additional tangent lines, so that circle A below will be inscribed in a triangle.

A

B

D

EC

Example 3

Use a compass and a straightedge to construct the lines tangent to circle C at point D.

C

D

continued

84



© Walch Education

Name: Date:

Example 4

Circle A and circle B are congruent. Construct a line tangent to both circle A and circle B.

A

B

85



6.3.1

Name: Date:

Problem-Based Task 6.3.1: Designing a T-Shirt LogoA local T-shirt shop is designing a logo for a company. The company’s logo is a circle inscribed in a triangle. The sketch below is incomplete. The designer needs to draw lines tangent to the circle from point C to point A and from point C to point B.

The designer also needs to calculate the perimeter of the triangle to be sure the logo fits in the allotted space. He knows the following information about the dimensions of the design:

• The diameter of circle G is 8 cm.

• CI = 4 cm

• CI , BH , and AJ are all congruent.

G

EA B

J H

D F

I

C

The designer needs to draw lines

tangent to the circle from point C to point A and from point C to

point B.

87



6.3.1

Name: Date:

continued

Use your knowledge of constructions to complete each problem that follows.

1. Construct circle A given point B on the circle. Construct a line tangent to circle A at point B.

2. Construct circle C given a point D not on the circle. Construct a line tangent to circle C through point D.

3. Construct two non-intersecting circles with congruent radii. Construct one common exterior tangent.

4. � ��AB is tangent to circle D at point B. Point A is an exterior point to circle D. What was the first

step that had to be completed to construct tangent � ��AB ?

5. � ��QR is tangent to circle S at point R. Between which two points must the sharp point of the compass have been placed in order to make the arc that formed point Q?

S

Q

R

Practice 6.3.1: Constructing Tangent Lines

89



6.3.1

Name: Date:

6. BC is tangent to circle A at point B. Describe the steps that were taken to construct the tangent.

A

D

B

E

C

7. Your friend says that if two lines are tangent to the same circle, they must intersect. Is your friend correct? Why or why not?

8. Construct circle A. Place three points on circle A and construct three tangent lines so the circle is inscribed in the triangle.

9. AB and AC are tangent to circle L in the diagram below. How can you use construction tools to verify that the segments are congruent?

LA

B

C

10. AB is tangent to circle C at point B. What can you conclude about the construction of point A?

90

Notes

Name: Date:

91

Notes

Name: Date:

92

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESLesson 4: Finding Arc Lengths and Areas of Sectors


6.4.1

Name: Date:

A child is riding a bicycle around a cul-de-sac. The cul-de-sac is circular with a diameter of 60 feet.

1. If the child rides her bicycle the entire circumference of the circle, how far will she have traveled?

2. If the child gets tired halfway through the ride and stops, how far will she have traveled?

Lesson 6.4.1: Defining Radians

Warm-Up 6.4.1

93

Unit 6 • CirCles with and without CoordinatesLesson 4: Finding Arc Lengths and Areas of Sectors



Name: Date:


Convert 40º to radians.

1. Set up a proportion.

2. Multiply both sides by π to solve for x.

continued

95



© Walch Education

Name: Date:

Example 2

Convert 3

4

π radians to degrees.

Example 3

A circle has a radius of 4 units. Find the radian measure of a central angle that intercepts an arc of length 10.8 units.

10.8 units

4 units

Example 4

A circle has a radius of 3.8 units. Find the length of an arc intercepted by a central angle measuring 2.1 radians.

2.1

3.8 units

Example 5

A circle has a diameter of 20 feet. Find the length of an arc intercepted by a central angle measuring 36º.

36°d = 20 feet

96



6.4.1

Name: Date:

Problem-Based Task 6.4.1: Around the Merry-Go-RoundA merry-go-round has a circumference of 160 feet and revolves at a speed of 6 miles per hour. How many feet does a carousel horse on the outer edge of the merry-go-round travel in a 2-minute ride? How many radians does a carousel horse travel in a 2-minute ride?

How many feet does a carousel

horse on the outer edge of the merry-go-round travel in a 2-minute ride? How many radians does a carousel horse

travel in a 2-minute ride?

97



6.4.1

Name: Date:

Use your knowledge of radian measures to complete the following problems.

1. Convert 80º to radians. Leave the answer in terms of π.

2. Convert 5

6

π radians to degrees.

3. A circle has a radius of 5 units. Find the radian measure of a central angle that intercepts an arc length of 15 units.

4. A circle has a radius of 18 units. Find the length of an arc intercepted by a central angle measuring 2.9 radians.

5. A circle has a radius of 11 units. Find the length of an arc intercepted by a central angle measuring 72º.

6. A central angle of 5

2

π radians intercepts an arc length of 46 units. What is the radius of the

circle, rounded to the nearest hundredth?

7. A standard dartboard has a radius of 170 mm and is split into 20 equal sections. What is the arc length of a single section on a dartboard rounded to the nearest millimeter?

8. How many radians does the hour hand on a clock travel through from 12 to 5?

9. A 26-inch diameter bicycle tire rotates 500 times. How many feet does the bicycle travel?

10. What is the difference between arc measure and arc length?

Practice 6.4.1: Defining Radians

99

Notes

Name: Date:

101

Notes

Name: Date:

102



6.4.2

Name: Date:

A new, decorative stained-glass window is in the shape of a semicircle with a 2-foot diameter. The window is to be covered with fabric and outlined with ribbon until an unveiling ceremony.

1. How many feet of ribbon will be needed if you need 2 extra feet of ribbon to tie a bow?

2. How many square feet of fabric will be needed?

Lesson 6.4.2: Deriving the Formula for the Area of a Sector

Warm-Up 6.4.2

103



© Walch Education

Name: Date:


A circle has a radius of 24 units. Find the area of a sector with a central angle of 30º.

30°24 units

1. Find the area of the circle.

2. Set up a proportion.

3. Multiply both sides by the area of the circle to find the area of the sector.

continued

105




Name: Date:

Example 2

A circle has a radius of 8 units. Find the area of a sector with a central angle of 3

4

π radians.

8 units

3 4

Example 3

A circle has a radius of 6 units. Find the area of a sector with an arc length of 9 units.

6 units

9 units

106



6.4.2

Name: Date:

Problem-Based Task 6.4.2: Pizza SpecialsA pizza parlor has a $5 lunch special this week for 3 pieces of a small cheese pizza and a soft drink. Next week, the $5 special will be for 2 pieces of a large cheese pizza and a soft drink. A small pizza measures 10 inches in diameter and is cut into 6 equal slices. A large pizza measures 14 inches in diameter and is cut into 8 equal slices. Which special is the better deal?

Which special is the better deal?

107



6.4.2

Name: Date:

Use your knowledge of the areas of sectors to complete the following problems.

1. Find the area of a sector with a central angle of 9.6 radians and a radius of 21.4 units.

2. Find the area of a sector with a central angle of π3

radians and a radius of 12 units.

3. Find the area of a sector with a central angle of 44º and a radius of 56 units.

4. A circle has a radius of 8 units. Find the area of a sector with an arc length of 6 units.

5. A circle has a radius of 2 units. Find the arc length of a sector with an area of 12 square units.

6. A sector has a central angle of π2

radians and an area of 13 square units. What is the area of

the circle?

7. A personal pizza with a 6-inch diameter is cut into slices with a central angle of π2

radians.

What is the area of each slice?

8. A blueberry pie is made in a 101

4-inch diameter pie pan. The pie is cut into 8 equal slices. What

is the area of 1 slice?

9. A rotating sprinkler sprays a stream of water 40 feet long. The sprinkler rotates 190º. What is the area of the portion of the yard that is watered by the sprinkler?

10. A lighthouse projects a beam of light that can be seen from up to 4 miles away and covers an angle of 35º. What is the area of the region from which a ship can see the light from the lighthouse?

Practice 6.4.2: Deriving the Formula for the Area of a Sector

109

Notes

Name: Date:

111

Notes

Name: Date:

112

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESLesson 5: Explaining and Applying Area and Volume Formulas


6.5.1

Name: Date:

An athletic field is in the shape of a rectangle with a semicircle on each end. The field is 180 meters long and 60 meters wide. The amount of work needed to maintain the field depends on its total area. What is the total area of the field?

1. Draw a diagram of the field.

2. How can you break the shape of the field into separate pieces?

3. What area formulas do you need to use to find the total area of the field?

4. What is the radius of the circle? What is the length of the rectangle?

5. What is the total area of the field?

Lesson 6.5.1: Circumference and Area of a Circle

Warm-Up 6.5.1

113

Unit 6 • CirCles with and without CoordinatesLesson 5: Explaining and Applying Area and Volume Formulas



Name: Date:


Show how the perimeter of a hexagon can be used to find an estimate for the circumference of a circle that has a radius of 5 meters. Compare the estimate with the circle’s perimeter found by using the formula C = 2πr.

1. Draw a circle and inscribe a regular hexagon in the circle. Find the length of one side of the hexagon and multiply that length by 6 to find the hexagon’s perimeter.

2. Create a triangle with a vertex at the center of the circle. Draw two line segments from the center of the circle to vertices that are next to each other on the hexagon.

3. To find the length of BC , first determine the known lengths of PB and PC .

4. Determine m CPB∠ .

5. Use trigonometry to find the length of BC .

6. Determine m BPD∠ .

7. Use trigonometry to find the length of BD and multiply that value by 2 to find the length of BC .

8. Find the perimeter of the hexagon.

9. Compare the estimate with the calculated circumference of the circle.

continued

115



© Walch Education

Name: Date:

Example 2

Show how the area of a hexagon can be used to find an estimate for the area of a circle that has a radius of 5 meters. Compare the estimate with the circle’s area found by using the formula A = π r 2.

Example 3

Find the area of a circle that has a circumference of 100 meters.

Example 4

What is the circumference of a circle that has an area of 1,000 m2?

116



6.5.1

Name: Date:

Problem-Based Task 6.5.1: Designing a TableclothA factory is printing a large square design on a circular tablecloth. The square design should be as large as possible. The tablecloth has an area of 30 square feet. What should be the maximum side length of the square design? What area of the tablecloth will not be printed with the square design?

What should be the maximum side

length of the square design?

What area of the tablecloth will not be printed with the

square design?

117



6.5.1

Name: Date:

continued

Use your knowledge of circumference and area to complete each problem.

1. A circle has a regular octagon inscribed in it. The circle has a radius of 4 meters. Find the perimeter of the octagon. Use the formula 2π r to find the circumference of the circle. Why is the circumference found by using the formula a different length than the perimeter of the octagon?

2. A circle has a regular dodecagon (12-sided polygon) inscribed in it. The circle has a radius of 4 meters. Find the perimeter of the dodecagon. Then, find the circumference of the circle using 2π r. Why is the circumference found by using the formula a different length than the perimeter of the dodecagon?

3. Compare the results of problems 1 and 2. Which dissection is a better approximation of the circumference of the circle? Use a 15-sided regular polygon as the inscribed figure in a circle that has a radius of 4 meters. Calculate the polygon’s perimeter and compare it with the circle’s circumference.

4. A circle has a regular octagon inscribed in it. The circle has a radius of 4 meters. Find the area of the circle using the formula A = π r 2, then find the area of the octagon. Why is the area of the circle different from the area of the octagon?

5. A circle has a regular dodecagon inscribed in it. The circle has a radius of 4 meters. Find the area of the circle and then of the dodecagon. Why is the area of the circle different from the area of the dodecagon?

Practice 6.5.1: Circumference and Area of a Circle

119



6.5.1

Name: Date:

6. Compare the results of problems 4 and 5. Which dissection is a better approximation of the area of the circle? Use a 36-sided regular polygon as the inscribed figure in a circle that has a radius of 4 meters. Calculate the polygon’s area and compare it with the circle’s area.

7. A round dining room table has a wood top with a circumference of 32 feet. A woodworker is refinishing the top. He needs to find the area of the top to buy materials and know how long the job will take. What is the area of the tabletop?

8. A pizza has a circumference of 40 inches. What is the area of the pizza?

9. A carpenter is installing curved wood trim around a circular window. The window is a circle that has an area of 50 square feet. How many feet of wood trim are needed to surround the window? Measure the trim based on the length of the wood next to the window.

10. An artist paints a large blue circle on a yellow wall. The area of the painted circle is 150 square feet. What is the circumference of the circle?

120

Notes

Name: Date:

121

Notes

Name: Date:

122



6.5.2

Name: Date:

A gardener in Texas is designing a rainwater collection system. She will collect rainwater from her roof to water her garden. The collection barrels are cylinders that are 1.3 meters high and have a radius of 40 cm. A cylinder is a solid or hollow object that has two parallel bases connected by a curved surface. The gardener wants to collect at least 5 cubic meters of water.

1. In what way does a cylinder resemble the shape of a prism?

2. How are the formulas for the volumes of prisms and cylinders similar and different?

3. What do the formulas look like for the volumes of a prism and cylinder?

4. How many collection barrels does the gardener need?

Lesson 6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

Warm-Up 6.5.2

123




Name: Date:


Find the dimensions for a cylinder that has the same volume as a square prism with a base area of 9 square meters. The cylinder and the square prism should both have heights of 5 meters.

5 m5 m

1. Determine the relationship between two objects with the same volume.

2. Set up the formulas for the area of the base of the cylinder and the area of the base of the prism so that they are equal.

3. Solve the equation for r.

4. Calculate the volume for each object.

5. Verify that the two objects will have the same area at a height of 1 meter or any other height.

continued

125



© Walch Education

Name: Date:

Example 2

Find the dimensions for a cone that has the same volume as a pyramid of the same height as the cone. Both the cone and the pyramid have a height of 2 meters. The volume of the pyramid is 3 cubic meters. A cone and a pyramid both taper to a point or vertex at the top. The “slant” of the taper is linear, meaning it is a straight line. The dimensions of both the cone and the pyramid change at a constant rate from base to tip.

2 m2 m

Example 3

A new art museum is being built in the shape of a square pyramid. The height will be 50 meters. The art museum needs 86,400 cubic meters of space inside. What should be the side lengths of the base of the pyramid?

Example 4

Weston has two round balloons. One balloon has a radius that is 3 times the radius of the other balloon. How much more air will the larger balloon need than the smaller balloon?

Example 5

A teenager buying some chewing gum is comparing packages of gum in order to get the most gum possible. Each package costs the same amount. Package 1 has 20 pieces of gum shaped like spheres. Each piece has a radius of 5 mm. Package 2 has 5 pieces of gum shaped like spheres. Each piece has a radius of 10 mm. Which package should the teenager buy? Round to the nearest millimeter.

126



6.5.2

Name: Date:

Problem-Based Task 6.5.2: Cylinders of SandThe manager of road maintenance for a large city wants to reduce the amount of space needed to store piles of sand. The sand is used on icy roads in the winter. There are 3 piles of sand that are cone-shaped. Each pile has a circumference of 157 feet and a height of 20 feet. The manager is thinking of building a storage area that is in the shape of a cylinder. The cylinder will have the same height as the cone-shaped sand piles, and the cylinder’s circumference will be the same as that of the bases of the sand piles. How much area will be saved by building the new cylindrical container?

How much area will be saved

by building the new cylindrical

container?

127



6.5.2

Name: Date:

continued

Use your knowledge of volume to complete each problem. Round each answer to the nearest whole number.

1. A gasoline fuel storage tank at an oil refinery is a cylinder with a radius of 20 meters and a height of 10 meters. How many gallons of gasoline will the tank hold? There are 264.172 gallons in 1 cubic meter.

2. A storage container is a cylinder with a height of 28.2 cm and a radius of 5 cm. How many liters of water will the storage container hold? 1 liter = 1000 cm3.

3. A company makes candles in the shape of cones. Their best-selling candle has a height of 6 inches and a circumference of 12 inches. What volume of wax is needed to make 1 candle?

4. For problem 3 above, what would the base side lengths be for a square pyramid candle that has the same volume and height as the cone?

5. In July 2012, an ice cream company in England set a new world’s record for the largest ice cream cone ever made. The total height was 13 feet including the ice cream on top. The cone itself was approximately 9 feet tall and had a diameter of about 3.5 feet. How many gallons of ice cream were needed to fill just the cone part (not including the ice cream on top)? 1 cubic foot = 7.40852 gallons.

Practice 6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

129



6.5.2

Name: Date:

6. A small city wants to build a cylindrical water tank that holds 10,000,000 liters. The city wants the height of the tank to be 10 meters. What will be the tank’s diameter? 1 cubic meter = 1000 liters.

7. A history museum is building an outdoor model of a Mesoamerican pyramid located in South America. The outside of the pyramid will be made of small square stones. The inside will be concrete. The concrete part of the square pyramid will have side lengths of 5 feet and a height of 3 feet. What volume of concrete is needed to make the inside?

Use your knowledge of volume to complete each problem. Round each answer to the nearest hundredth.

8. The diameter of Mars is approximately 6,794 kilometers. What is the volume of Mars to the nearest cubic kilometer?

9. If the radius of a sphere is doubled, by how much does the volume of the sphere increase?

10. A manufacturer is shipping a spherical globe that fits exactly in a box shaped like a cube. The globe is touching all six sides of the box. If the volume of the box is 343 in3, what is the volume of the globe?

130

Notes

Name: Date:

131

Notes

Name: Date:

132

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESLesson 6: Deriving Equations


6.6.1

Name: Date:

A video game designer created the following diagram of a target.

– 20 – 10 10 20

15

10

5

– 5

– 10

– 15

A (4, 3)B (–12, 5)

x

y

1. What is the radius of the circle that contains point A? (Hint: Draw a right triangle and use the Pythagorean Theorem.)

2. What is the radius of the circle that contains point B?

Lesson 6.6.1: Deriving the Equation of a Circle

Warm-Up 6.6.1

133

Unit 6 • CirCles with and without CoordinatesLesson 6: Deriving Equations


© Walch Education

Name: Date:


Derive the standard equation of the circle with center (0, 0) and radius 5.

1. Sketch the circle.

y

x10–10

–10

10

8

6

2

–2

–4

–6

–8

–8 –6 –4 –2 2 4

4

6 80

2. Use the Pythagorean Theorem to derive the standard equation.

continued

135




Name: Date:

Example 2

Derive the standard equation of the circle with center (2, 1) and radius 4. Then use a graphing calculator to graph your equation.

Example 3

Write the standard equation and the general equation of the circle that has center (–1, 3) and passes through (–5, 5).

Example 4

Find the center and radius of the circle described by the equation x2 + y2 – 8x + 2y + 2 = 0.

Example 5

Find the center and radius of the circle described by the equation 4x2 + 4y2 + 20x – 40y + 116 = 0.

136



6.6.1

Name: Date:

Problem-Based Task 6.6.1: Nurturing an InvestmentAnna’s landscaping company has a contract to improve and maintain a municipal park. Anna made a scale drawing of the park on a coordinate system, using meters as the unit of distance. She has already installed two permanent sprinkler outlets. Sprinkler 1 waters inside the region whose boundary has the equation x2 + y2 – 20x – 20y + 136 = 0. Sprinkler 2 waters inside the region whose boundary has the equation x2 + y2 – 50x – 24y + 669 = 0. Anna bought an expensive tree and she wants to plant it at the point (17, 8), where she thinks it will be watered by both sprinklers. Will the tree be watered by both sprinklers at that point? Draw a sketch that illustrates your answer.

Will the tree be watered by both

sprinklers at that point? Draw a sketch that

illustrates your answer.

137



6.6.1

Name: Date:

For problems 1–4, write the standard equation of the circle described.

1. The center is (0, 0) and the radius is 9.

2. The center is (–5, 1) and the radius is 4.

3. The center is (0, –3.2) and the radius is 2.2.

4. The center is (–2, 2) and the circle passes through (1, –3).

Use the provided information in each problem that follows to solve.

5. Write the general equation of the circle with center (5, 0) and radius 3 2 .

6. Find the center and radius of the circle described by the equation x2 + y2 + 6x – 4y – 27 = 0.

7. Find the center and radius of the circle described by the equation 2 2 2 69

202 2x y x y+ + − + = .

8. A particular cell phone tower is designed to service a 12-mile radius. The tower is located at (–3, 5) on a coordinate plane whose units represent miles. What is the standard equation of the outer boundary of the region serviced by the tower? Is a cell phone user at (8, 0) within the service range? Explain.

9. A pizza restaurant will deliver up to 5 miles. The restaurant is located at the origin on a coordinate plane whose units represent miles. What is the standard equation of the outer boundary of the delivery region? Customers are located at A (4, 3), B (5, 0), and C 2 21,( ) . Which of these customers, if any, are on the outer boundary? Explain.

10. Marco is a park ranger stationed in a fire tower. The tower is on a coordinate plane whose units represent miles. Marco is responsible for monitoring a region whose boundary has the equation x2 + y2 + 3x + y – 6.5 = 0. What is the geometric description of the region for which Marco is responsible?

Practice 6.6.1: Deriving the Equation of a Circle

139

Notes

Name: Date:

141

Notes

Name: Date:

142



6.6.2

Name: Date:

The diagram below shows a window on a coordinate plane with feet as the unit of distance. The width

of the window is 4 feet. Points B, C, and D follow the equation y x= − +1

462 .

y

C

DB

EAx

1. What are the coordinates of point C? Explain.

2. What are the coordinates of points A, B, D, and E? Explain.

Lesson 6.6.2: Deriving the Equation of a Parabola

Warm-Up 6.6.2

143



© Walch Education

Name: Date:


Derive the standard equation of the parabola with focus (0, 2) and directrix y = –2 from the definition of a parabola. Then write the equation by substituting the vertex coordinates and the value of p directly into the standard form.

1. To derive the equation, begin by plotting the focus. Label it F (0, 2). Graph the directrix and label it y = –2. Sketch the parabola. Label the vertex V.

2. Let A (x, y) be any point on the parabola.

3. Apply the definition of a parabola to derive the standard equation using the distance formula.

4. To write the equation using the standard form, first determine the coordinates of the vertex and the value of p.

5. Use the results found in step 4 to write the equation.

continued

145




Name: Date:

Example 2

Derive the standard equation of the parabola with focus (–1, 2) and directrix x = 7 from the definition of a parabola. Then write the equation by substituting the vertex coordinates and the value of p directly into the standard form.

Example 3

Derive the standard equation of the parabola with focus (0, p) and directrix y = –p, where p is any real number other than 0.

Example 4

Write the standard equation of the parabola with focus (–5, –6) and directrix y = 3.4. Then use a graphing calculator to graph your equation.

continued

146



© Walch Education

Name: Date:

Example 5

The following diagram shows a plan for a top view of a stage. The back wall is to be on a parabolic curve from A to B so that all sound waves coming from point F that hit the wall are redirected in parallel paths toward the audience. F is the focus of the parabola and V is the vertex.

B

A

V F

An engineer draws the parabola on a coordinate plane, using feet as the unit of distance. The focus is (–7, 0), the directrix is x = –25, and points A and B are on the y-axis. What is the equation of the parabola? What is the width of the stage, AB?

B (0, y2)

A (0, y1)

V

x = –25

F

(–7, 0)

y

x

147



6.6.2

Name: Date:

Problem-Based Task 6.6.2: A Ball in FlightA softball player hits the ball when it is directly above home plate. The ball then follows a parabolic path until it hits the ground. The path of the ball is represented by a portion of a parabola on a coordinate plane, where the origin represents home plate and the unit of distance is feet. The focus of the parabola is (100, –18.5) and the directrix is y = 106.5.

Write the equation of the path of the ball and supply the following key facts: the maximum height of the ball, the height of the ball when it is hit by the bat, and how far the ball travels horizontally before it hits the ground.

Write the equation of the path of

the ball.

149



6.6.2

Name: Date:

continued

For problems 1–4, derive the standard equation of the parabola with the given focus and directrix. Also, write the equation that shows how you applied the distance formula.

1. focus: (0, 5); directrix: y = –5

2. focus: (–2, 0); directrix: x = 2

3. focus: (–3, 5); directrix: y = –3

4. focus: (0, –3); directrix: x = 8

For problems 5 and 6, write the standard equation of the parabola with the given focus and directrix.

5. focus: (4.5, –5.5); directrix: y = 1.5

6. focus: (–10, 20); directrix: x = 110

Use what you know about parabolas to solve problems 7–10.

7. Identify the vertex, focus, and directrix of the parabola whose equation is (x – 3)2 = 10(y + 1).

8. The diagram below shows a parabolic satellite dish antenna. Incoming TV signals reflect off of the dish and toward the feed, or receiver, of the antenna, located at point F. A cross section of the dish is a section of a parabola.

F

Cross-section view

The parabola is placed on a coordinate plane whose unit of distance is feet. The focus F is (0, 3.25) and the directrix is y = –3.25. What is the standard equation of the parabola?

Practice 6.6.2: Deriving the Equation of a Parabola

151



6.6.2

Name: Date:

9. The diagram below shows a parabolic arch bridge.

Road

x-axisA B

Height

An engineer draws the bridge on a coordinate plane so that points A and B are on the x-axis, the focus is (60, –5.25), and the directrix is y = 55.25. The unit of distance on the coordinate plane is feet. What is the standard equation of the parabola? What is the height of the bridge? What is the distance AB? Sketch the parabola, showing the coordinates of A, B, and the vertex.

10. The diagram below shows a radio telescope dish. Incoming light rays reflect off of the dish and toward the feed, located at point F. A cross section of the dish is a section of a parabola. The feed is 40 inches above the vertex. The diameter of the dish at the top is 8 feet.

F

Cross-section view

Depth

A technician draws the parabola on a coordinate plane so that the vertex is at the origin. What is the equation of the parabola on the coordinate plane? What is the depth of the dish?

152

Notes

Name: Date:

153

Notes

Name: Date:

154

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESLesson 7: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas


6.7.1

Name: Date:

Juan is planning to make a circular tabletop by joining wooden planks. He draws the following diagram of the tabletop on a coordinate system in which distance is represented in feet. The line segments represent the edges of the planks, which are all parallel.

P

A (6, 7)

B (3, 1)

Q

y

x

1. What is the length of line segment AB ? Show your work.

2. What is the slope of line segment PQ? Explain.

Lesson 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and ParabolasWarm-Up 6.7.1

155

Unit 6 • CirCles with and without CoordinatesLesson 7: Using Coordinates to Prove Geometric Theorems About Circles and



Name: Date:


Given the point A (–6, 0), prove or disprove that point A is on the circle centered at the origin and passing through − −( )2 4 2, .

1. Draw a circle on a coordinate plane using the given information.

2. Find the radius of the circle using the distance formula.

3. Find the distance of point A from the center P to determine whether it is on the circle.

continued

157

Unit 6 • CirCles with and without CoordinatesLesson 7: Using Coordinates to Prove Geometric Theorems About Circles and


© Walch Education

Name: Date:

Example 2

Prove or disprove that the quadratic function graph with vertex (–4, 0) and passing through (0, 8) has its focus at (–4, 1).

Example 3

The following information is given about a parabola:

• The vertex V is at (0, 0).

• The focus F is at (p, 0), with p > 0.

• The line segment through F is perpendicular to the axis of symmetry and connects two points of the parabola.

Prove that the line segment through F has length 4p.

Example 4

Prove or disprove that the points A (4, 2), B (–2, 5), C (6, 5), and D (–4, 10) are all on the quadratic function graph with vertex V (2, 1) that passes through E (0, 2).

Example 5

Prove or disprove that P (–2, 1), Q (6, 5), R (8, 1), and S (0, –3) are vertices of a rectangle that is inscribed in the circle centered at C (3, 1) and passing through A 1 1 21, +( ) .

158



6.7.1

Name: Date:

Problem-Based Task 6.7.1: A Circle Graph for LunchAlena conducted a survey of her classmates’ lunch preferences. She drew a circle graph on a transparency sheet to show the results. She placed the circle graph on a coordinate plane and positioned it so that points R and T have the coordinates R (24, 22) and T (8, 10). Alena claims that the coordinates of point P are (8, 22). Is she correct? Prove or disprove her claim.

P

Q

R (24, 22)

T (8, 10)

C

Lunch Preferences

Tuna 10%

Cold cut15%

Chicken20%

Pizza 30%

Burger25%

S

Alena claims that the coordinates of point P are (8, 22). Is she correct? Prove or disprove

her claim.

159



6.7.1

Name: Date:

continued

Use the given information to prove or disprove each statement. Justify your reasoning.

1. Prove or disprove that point P 2 2 3,( ) lies on the circle centered at the origin R and passing through the point Q (4, 0).

2. Prove or disprove that point Q 6 2 5,( ) is on the circle centered at the origin A and passing through the point B −( )3 55, .

3. Given the points A (–2, 4), B (1, 1), and C (3, 9), prove or disprove that these points are on the

parabola with focus F 01

4,

and directrix y= −1

4.

4. Given the points A (–2, 4), B (–1, 1), C (2, 4), and D (3, 9), prove or disprove that points are on the quadratic function graph with vertex V (0, 0) and passing through E (5, 25).

5. Prove or disprove that the points P (8, 18), Q (–4, 18), R (6, 8), and S (–2, 8) are all on the quadratic function graph with vertex V (2, 0) that passes through T (0, 2).

Practice 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

161



6.7.1

Name: Date:

6. Prove or disprove that the point P (–5, 4) lies on the circle centered at C (–1, 2) and passing through the point D (1, –2).

7. Prove or disprove that the points A (8, 6), B (8, –6), and C (–10, 0) are the vertices of an isosceles triangle inscribed in the circle centered at the origin Q and passing through the point P −( )3 91, .

8. The diagram below shows a target at a carnival dart game. The diagram is on a coordinate system. A player wins a prize by hitting a shaded ear. The shaded ears are circles as follows:

• a circle centered at C1 (10, 50) and passing through P

1 (13, 54)

• a circle centered at C2 (30, 50) and passing through P

2 (26, 47)

Bradley throws two darts, hitting the points A (14, 46) and B (26, 55). Does he win a prize? Justify your answer.

y

x

continued

162



6.7.1

Name: Date:

9. A graphic artist created the sketch of a baseball shown below, using the following graphs on a coordinate system:

• a parabola with vertex V1 (1, 0) and focus F

1 (3, 0)

• a parabola with vertex V2 (–1, 0) and focus F

2 (–3, 0)

• a circle with center R (0, 0) and passing through the point P 2 21,( )Prove or disprove that the coordinates of the four labeled points are as follows: A (–3, 4), B (3, 4), C (–3, –4), and D (3, –4).

C

A

D

B

y

x

10. The following graph represents the parabolic path of a baseball. A batter hit the ball when the ball was 3 feet above the ground. After being hit, the ball reached a maximum height of 75 feet, which occurred when it had traveled 120 feet horizontally. Each unit on the graph represents 1 foot. Prove or disprove that the ball was 43 feet above the ground when it had traveled 200 feet horizontally.

(0, 3)

V (120, 75)

y

x

163

Notes

Name: Date:

165

Notes

Name: Date:

166

UNIT 6 • CIRCLES WITH AND WITHOUT COORDINATESStation Activities Set 1: Circumference, Angles, Arcs, Chords, and Inscribed Angles

© Walch Education CCSS IP Math II Teacher Resource U6-367

Name: Date:

continued

You will be given a plastic coffee can lid, a tape measure, a black marker, a compass, a ruler, and white paper.

1. As a group, use the black marker to mark a starting point on your coffee lid.

Roll the coffee can lid along the tape measure so you can measure the distance around the edge of the coffee can lid.

What is the mathematical name for this distance?

What is the distance around the edge of the lid in inches?

Repeat this measurement three more times to verify your answer.

2. Trace the coffee can lid on the white paper.

Use the ruler and compass to find the center of the circle.

3. What is the radius of the circle?

What is the diameter of the circle?

How does the radius relate to the diameter?

4. What is π ?

Station 1

167


CCSS IP Math II Teacher Resource © Walch Education U6-368

Name: Date:

5. What is π times twice the radius of your circle?

Does this match your answer in problem 1? Why or why not?

6. What is π times the diameter of your circle?

Does this match your answer in problem 1? Why or why not?

7. Do your answers for problems 5 and 6 match? Why or why not?

8. Based on your observations in problems 1–7, what is the formula for the circumference of a circle written in terms of the radius?

What is the formula for the circumference of a circle written in terms of the diameter?

continued

168


© Walch Education CCSS IP Math II Teacher Resource U6-369

Name: Date:

9. Larry installed a circular pool in his backyard. The pool has a diameter of 20 feet. What is the circumference of the pool? Show your work in the space below.

10. Lisa is running for class president and passed out buttons that each have a circumference of 6.28 inches. What is the radius of each button? Show your work in the space below.

169

student workbook with scaffolded practice unit 6 · 2017. 8. 9. · given that all circles are...

Documents