© 2010 carnegie learning, inc. - oak park geometry classes€¦ · 11 chapter 11 | circles 633...

11

Chapter 11 | Circles 633

C H A P T E R

11 Circles

11.1 Riding a Ferris WheelIntroduction to Circles | p. 635

11.2 Take the WheelCentral Angles, Inscribed Angles,

and Intercepted Arcs | p. 643

11.3 Manhole Covers Measuring Angles Inside and Outside

of Circles | p. 655

11.4 Color TheoryChords | p. 669

11.5 Solar EclipsesTangents and Secants | p. 681

11.6 Replacement for a Carpenter’s SquareInscribed and Circumscribed Triangles

and Quadrilaterals | p. 691

11.7 GearsArc Length | p. 701

11.8 Playing DartsSectors and Segments of a Circle | p. 707

11.9 The Coordinate PlaneCircles and Polygons on the Coordinate

Plane | p. 713

© 2

010

Car

negi

e Le

arni

ng, I

nc.

Gears are circular objects used to transmit rotational forces from one

mechanical device to another. Often, gears are used to speed up or slow

down the rate of rotation of a mechanical object, such as a driveshaft or axle.

Interlocking gears of different sizes revolve at different rates, with a larger gear

completing a full revolution more slowly than a smaller one. You will use the arc

length of a circle to better understand how gears work together.

634 Chapter 11 | Circles

11

Introductory Problem for Chapter 11

That Darn Kitty!

A neighbor gave you a plate of cookies as a housewarming present. Before you

could eat a single cookie, the cat jumped onto the kitchen counter and knocked the

cookie plate onto the floor, shattering it into many pieces. The cookie plate will need

to be replaced and returned to the neighbor. Unfortunately, cookie plates come in

various sizes and you need to know the exact diameter of the broken plate.

It would be impossible to reassemble all of the broken pieces, but one large chunk

has remained intact as shown.

You think to yourself that there has to be an easy way to determine the diameter of the

broken plate. As you sit staring at the large piece of the broken plate, your sister Sarah

comes home from school. You update her on the latest crisis and she begins to smile.

Sarah tells you not to worry because she learned how to solve for the diameter of the

plate in geometry class today. She gets a piece of paper, a compass, a straightedge,

a ruler, and a marker out of her backpack and says, “Watch this!”

What does Sarah do? Describe how she can determine the diameter of the plate

with the broken piece. Then, show your work on the broken plate shown.

© 2

010

Car

negi

e Le

arni

ng, I

nc.

Lesson 11.1 | Riding a Ferris Wheel 635

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11OBJECTIVESIn this lesson you will:● Identify parts of a circle.● Draw parts of a circle.

KEY TERMS● center of a circle ● minor arc● chord ● semicircle● secant of a circle ● tangent of a circle ● point of tangency ● central angle● inscribed angle● arc● major arc

11.1

The first Ferris wheel was built in 1893 for the Chicago World’s Fair to rival the Eiffel

Tower, which was built for the Paris World’s Fair. Today, the Sky Dream Fukuoka

Ferris wheel in Japan is the world’s largest Ferris wheel.

Riding a Ferris WheelIntroduction to Circles

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 1 Going Around and Around A Ferris wheel is in the shape of a circle.

Recall, a circle is the set of all points in a plane that are equidistant from a given

point, which is called the center of the circle. The distance from a point on the

circle to the center is the radius of the circle. A circle is named by its center. For

instance, the circle seen in the Ferris wheel is circle P.

1. Use the circle to answer the questions.

a. Name the circle.

A

OE C

D

BF

Recall that a circle is a locus of points on a plane equidistant from a given point.

Because the points in a locus are infinite in number, they merge together and

appear to resemble a curve. If you used the world’s most powerful magnifying glass

and focused it on any portion of the curve, you would see millions and millions of

microscopic points. Only those points are considered to be the actual circle.

Points associated with a circle appear in one of three possible regions: points are

located in the interior of a circle, points are located on the circle, or points are

located in the exterior of a circle.

© 2

010

Car

negi

e Le

arni

ng, I

nc.

b. For each of the points associated with the circle shown, identify which

points are located in the interior of the circle, on the circle, or in the exterior

of the circle. Which point appears to be the center point of the circle?

c. Use a straightedge to draw ___

OB , the radius of circle O. Where are the

endpoints located with respect to the circle?

d. How many radii does a circle have? Explain your reasoning.

e. Use a straightedge to draw ___

AC . Then use a straightedge to draw ___

BD .

How are the line segments different? How are they the same?

Both line segments are chords of the circle “because the endpoints are on the

circle.” Segment AC is called a diameter of the circle.

f. Why is ___

BD not considered a diameter?

Take NoteRadii is the plural

of radius.

11



© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

g. How does the length of the diameter of a circle relate to the length of

the radius?

h. Are all radii of the same circle, or of congruent circles (always, sometimes,

or never), congruent? Explain.

A secant of a circle is a line that intersects a circle at exactly two points.

2. Draw a secant using the circle shown.

Z

3. Explain the difference between a chord and a secant.

4. Why is the diameter of a circle considered the longest chord in a circle?

A tangent of a circle is a line that intersects a circle at exactly one point. The point

of intersection is called the point of tangency.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

5. Draw a tangent using circle Z shown.

Z

6. Choose another point on the circle. How many lines can you draw through this

point tangent to the circle?

7. Explain the difference between a secant and a tangent.

8. Check the appropriate term(s) associated with each characteristic in the

table shown.

Characteristic Chord Secant Diameter Radius Tangent

A line

A line segment

A line segment having

both endpoints on the

circle

A line segment having one

endpoint on the circle

A line segment passing

through the center of the

circle

A line intersecting a circle

at exactly two points

A line intersecting a circle

at exactly one point


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 2 Sitting on the Wheel

A central angle is an angle of a circle whose vertex is the center of the circle.

An inscribed angle is an angle of a circle whose vertex is on the circle.

1. Four friends are riding a Ferris wheel in the positions shown.

Dru

Kelli

Marcus

Wesley

O

a. Draw a central angle where Dru and Marcus are located on the sides of

the angle.

b. Draw an inscribed angle where Kelli is the vertex and Dru and Marcus are

located on the sides of the angle.

c. Draw an inscribed angle where Wesley is the vertex and Dru and Marcus are

located on the sides of the angle.

d. Compare and contrast these three angles.

An arc of a circle is any unbroken part of the circumference of a circle. An arc is

named using its two endpoints. The symbol used to describe arc AB is ⁀ AB .

A major arc of a circle is the largest arc formed by a secant and a circle. It goes

more than half way around a circle.

A minor arc of a circle is the smallest arc formed by a secant and a circle.

It goes less than half way around a circle.

A semicircle is exactly half of a circle.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

2. A copy of the Ferris wheel from Question 1 is shown.

O

a. Label the location of each person with the first letter of his or her name.

b. Use your pencil to trace ⁀ DM . Describe what you traced.

c. Use your pencil to trace ⁀ MD . Describe what you traced.

d. What is the difference between ⁀ DM and ⁀ MD ?

e. Draw a diameter on the circle shown so that point D is an endpoint.

Label the second endpoint as point Z. The diameter divided the circle into

two semicircles.

f. Use your pencil to trace one semicircle. Describe what you traced.

g. Use your pencil to trace a different semicircle. Describe what you traced.

To avoid confusion, three points are used to name semicircles and major arcs.

The first point is an endpoint of the arc, the second point is any point at which the

arc passes through and the third point is the other endpoint of the arc.

h. Name each semicircle.

i. Name all minor arcs.

j. Name all major arcs.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 3 Name the Parts

Use the diagram shown to answer Questions 1 through 7.

OI

R

L

C

E

1. Name a diameter.

2. Name a radius.

3. Name a central angle.

4. Name an inscribed angle.

5. Name a minor arc.

6. Name a major arc.

7. Name a semicircle.

Be prepared to share your solutions and methods.

Lesson 11.2 | Take the Wheel 643

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Central Angles, Inscribed Angles, and Intercepted Arcs

OBJECTIVESIn this lesson you will:● Determine the measures of arcs.● Use the Arc Addition Postulate.● Determine the measures of central

angles and inscribed angles.● Prove the Inscribed Angle Theorem.● Prove the Parallel Lines-Congruent

Arcs Theorem.

KEY TERMS● degree measure (of an arc)● adjacent arcs● Arc Addition Postulate● intercepted arc● Inscribed Angle Theorem● Parallel Lines-Congruent Arcs Theorem

11.2

Before airbags were installed in car steering wheels, the recommended position for

holding the steering wheel was the 10-2 position. Now, one of the recommended

positions is the 9-3 position to account for the airbags. The numbers 10, 2, 9, and

3 refer to the numbers on a clock. So the 10-2 position means that one hand is at

10 o’clock and the other hand is at 2 o’clock.

Take the Wheel

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

The circles shown represent steering wheels, and the points on the circles represent

the positions of a person’s hands.

A

O

B

C D

P

Recall that the degree measure of a circle is 360°.

Each minor arc of a circle is associated with and determined by a specific central

angle. The degree measure of a minor arc is the same as the degree measure of

its central angle. For example, if the measure of central angle PRQ is 30° then the

degree measure of its minor arc PQ is equal to 30°. Using symbols, this can be

expressed as follows: If �PRQ is a central angle and m�PRQ � 30°, then

m ⁀ PQ � 30°.

1. For each circle, use the given points to draw a central angle. The hand position

on the left is 10-2 and the hand position on the right is 11-1. What are the

names of the central angles?

2. Without using a protractor, determine the central angle measures. Explain

your reasoning.

3. How do the measures of these angles compare?

4. Why do you think the hand position represented by the circle on the left

was recommended and the hand position represented on the right is

not recommended?

PROBLEM 1 Keep Both Hands on the Wheel


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

5. Describe the measures of the minor arcs you named in Question 3

using symbols.

6. Plot and label point Z on each circle so that it does not lie between the

endpoints of the minor arcs you identified in Question 5. Determine the

measures of the major arcs that have the same endpoints as the minor

arcs in Question 5. Explain your reasoning.

7. What is the measure of a semicircle? Explain your reasoning.

8. If the measures of two central angles of the same circle (or congruent circles)

are equal, are their corresponding minor arcs congruent? Explain.

9. If the measures of two minor arcs of the same circle (or congruent circles)

are equal, are their corresponding central angles congruent? Explain.

Adjacent arcs are two arcs of the same circle sharing a common endpoint.

10. Draw and label two adjacent arcs on circle O shown.

O


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

The Arc Addition Postulate states: “The measure of an arc formed by two adjacent

arcs is the sum of the measures of the two arcs.”

11. Apply the Arc Addition Postulate to the adjacent arcs you created for Question 10.

12. Name two other postulates you have studied that are similar to the Arc

Addition Postulate.

An intercepted arc is an arc associated with and determined by angles of the circle.

An intercepted arc is a portion of the circumference of the circle located on the

interior of the angle whose endpoints lie on the sides of an angle.

O

13. Draw inscribed �PSR on circle O.

14. Name the intercepted arc associated with �PSR.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

15. Consider the central angle that is shown. Use a straightedge to draw an

inscribed angle that contains points A and B on its sides. Name the vertex

of your angle point P.

A

B

O

a. What do the angles have in common?

b. Use your protractor to measure the central angle and the inscribed angle.

How is the measure of the inscribed angle related to the measure of the

central angle and the measure of ⁀ AB ?

c. Use a straightedge to draw a different inscribed angle that contains points

A and B on its sides. Name its vertex point Q. Measure the inscribed angle.



d. Use a straightedge to draw one more inscribed angle that contains points

A and B on its sides. Name its vertex point R. Measure the inscribed angle.



16. What can you conclude about inscribed angles that have the same

intercepted arc?


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

17. Create an Inscribed Angle Conjecture about the measure of an inscribed angle

and the measure of its intercepted arc.

18. Inscribed angles formed by two chords can be drawn three different ways with

respect to the center of the circle.

a. Case 1: Use circle O shown to draw and label inscribed �MPT such that the

center point lies on one side of the inscribed angle.

O

b. Case 2: Use circle O shown to draw and label inscribed �MPT such that the

center point lies on the interior of the inscribed angle.

O

c. Case 3: Use circle O shown to draw and label inscribed �MPT such that the

center point lies on the exterior of the inscribed angle.

O


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

19. To prove your Inscribed Angle Conjecture, you must prove each case in

Question 18, parts (a) through (c).

a. Use the diagram provided to complete the proof for Case 1.

OPM

T

Given: �MPT is inscribed in circle O.

m�MPT � x

Point O lies on diameter ____

PM .

Prove: m�MPT � 1 __ 2 m ⁀ MT

Statements Reasons

1. �MPT is inscribed in circle O.

m�MPT � x

Point O lies on diameter ___

PM .

1. Given

2. Connect points O and T to

form radius ___

OT .

2. Construction


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

b. Use the diagram provided to complete the proof for Case 2.

O

P

M

R

T


Point O is in the interior of �MPT.

m�MPO � x

m�TPO � y


Statements Reasons

1. �MPT is inscribed in circle O.

Point O is in the interior of �MPT.

m�MPO � x

m�TPO � y

1. Given

2. Construct diameter ___

PR . Connect

points O and T to form radius ___

OT .

Connect points O and M to form

radius ____

OM .

2. Construction


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

c. Use the diagram provided to complete the proof for Case 3.

Write a paragraph proof or create a two-column proof.

Hint: You will need to construct a diameter through point P and

construct radii ____

OM and ___

OT .

O

P

T

M

R


Point O is in the exterior of �MPT.


Statements Reasons


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Congratulations! You have just proven the Inscribed Angle Conjecture. It is now

known as the Inscribed Angle Theorem. You can now use this theorem as a valid

reason in proofs.

The Inscribed Angle Theorem states: “The measure of an inscribed angle is

one-half the measure of its intercepted arc.”

PROBLEM 2 Parallel Lines Intersecting a Circle

Allisa was excited because she invented her own conjecture in geometry class! The

focus of the lesson was the relationship between an inscribed angle and the measure

of its intercepted arc. Her teacher told the class to discover something they didn’t

already know using what they learned in class. Allisa was busy drawing different

kinds of diagrams and she decided to draw a circle with two parallel lines as shown.

A

O R

P

L

Allisa quickly wrote the following conjecture: “Parallel lines intercept congruent

arcs on a circle.” She named her conjecture the Parallel Lines-Congruent

Arcs Conjecture.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

1. Create a two-column proof of the Parallel Lines-Congruent Arcs Conjecture.

Given:

Prove:

Statements Reasons

Congratulations! You have just proven the Parallel Lines-Congruent Arcs Conjecture.

It is now known as the Parallel Lines-Congruent Arcs Theorem. You can now use

this theorem as a valid reason in proofs.

PROBLEM 3 Determine the Measure

____

MP is a diameter of circle O.

OP

WT

M

KS 1. If m ⁀ MT � 124° , determine m�TPW.

Explain your reasoning.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Use the diagram to answer Questions 2 through 4.

O

KJ

F G

2. Are radii ___

OJ , ___

OK , ___

OF , and ____

OG all congruent? Explain your reasoning.

3. Is m ⁀ FG greater than m ⁀ JK ? Explain your reasoning.

4. If m�FOG � 57°, determine m ⁀ JK and m ⁀ FG . Explain your reasoning.

5. DeJaun told Thomas there was not enough information

A

B

to determine whether circle A was congruent to circle B.

He said they would have to know the length of a radius

in each circle to determine whether the circles were

congruent. Thomas explained to DeJaun why he was

incorrect. What did Thomas say to DeJaun?


Lesson 11.3 | Manhole Covers 655

© 2

010

Car

negi

e Le

arni

ng, I

nc.

OBJECTIVESIn this lesson you will:● Determine measures of angles formed

by two chords.

● Determine measures of angles formed by two secants.

● Determine measures of angles formed by a tangent and a secant.

● Determine measures of angles formed by two tangents.

● Prove the Interior Angles of a

Circle Theorem.

● Prove the Exterior Angles of a

Circle Theorem.

KEY TERMS● Interior Angles of a

Circle Theorem● Exterior Angles of a

Circle Theorem

11.3

Manhole covers are heavy removable plates that are used to cover maintenance

holes in the ground. Most manhole covers are circular and can be found all over

the world. The tops of these covers can be plain or have beautiful designs cast

into their tops.

Manhole CoversMeasuring Angles Inside and Outside of Circles

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 1 Inside the Circle

Circle O shows a simple manhole cover design.

m ⁀ BD � 70°

m ⁀ AC � 110°

A B

C

D

E

O

1. Consider �BED. How is this angle different from the angles that you have seen

so far in this chapter? How is this angle the same?

2. Can you determine the measure of �BED with the information you have so far?

If so, how? Explain your reasoning.

3. Draw chord ___

CD . Use the information given in the figure to name the measures

of any angles that you do know. Explain your reasoning.

4. How does �BED relate to �CED?

5. Write a statement showing the relationship between m�BED, m�EDC,

and m�ECD.

6. What is the measure of �BED?

7. Consider circle P shown. Draw chord ___

XY on the figure shown.

V

X

W

Y

P

Z

a. Write a statement for m�WXY in terms of m ⁀ WY .

b. Write a statement m�VYX for in terms of m ⁀ VX .


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

c. Write a statement for m�WZY in terms of m ⁀ WY and m ⁀ VX .

The measure of an angle formed by two intersecting chords is half of the sum

of the measures of the arcs intercepted by the angle and its vertical angle.

d. Consider �WZY again. What is the arc that is intercepted by �WZY?

e. Name the angle that is vertical to �WZY. Then name the arc that

is intercepted by the angle vertical to �WZY.

8. Consider the figure. By the statement you wrote previously, you can state

that �CEB � 1 __ 2 (m ⁀ CB � m ⁀ AD ). Write similar statements for m�AED, �AEC,

and �DEB.

O

E

A

B

C

D

The Interior Angles of a Circle Theorem states: “If an angle is formed by

two intersecting chords or secants of a circle such that the vertex of the angle is

in the interior of the circle, then the measure of the angle is half of the sum of the

measures of the arcs intercepted by the angle and its vertical angle.”


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

9. Create a two-column proof of the Interior E

F

O

G

K

H

Angles of a Circle Theorem.

Given: Chords ___

EK and ____

GH intersect at point F in

circle O.

Prove: m�KFH � 1 __ 2 (m ⁀ HK � m ⁀ EG )

Statements Reasons

1. Chords ___

EK and ____

GH intersect at point

F in circle O.

1. Given

2. Connect points E and H to form

chord ___

EH .

2. Construction

Circle T shows another simple manhole cover design.

m ⁀ KM � 80°

m ⁀ LN � 30°

KL

M

NT

1. Consider ___

KL and ____

MN . Use a straightedge to draw secants that coincide with each

segment. Where do the secants intersect? Label this point as point P on the figure.

2. Draw chord ___

KN . Can you determine the measure of �KPM with the information

you have so far? If so, how? Explain your reasoning.

PROBLEM 2 Outside the Circle


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

3. Use the information given in the figure for Problem 2 to name the measures of

any angles that you do know. Explain how you determined your answers.

4. How does �KPN relate to �KPN?

5. Write a statement showing the relationship between m�KPN, m�NKP,

and m�KNM.

6. What is the measure of �KPN?

7. Describe the measure of �KPM in terms of the measures of both arcs

intercepted by �KPM.

If an angle is formed by two intersecting secants so that the angle is outside the

circle, then the measure of this angle is half of the difference of the arc measures

that are intercepted by the angle.

8. Consider the figure. Use the previous statement to write a statement for

m�CAD in terms of the arc measures that are intercepted by �CAD.

A B C

D

E O


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

9. In circle P, the line through point T is tangent to the circle and is perpendicular

to ___

UT .

U

X P Y

TV W

a. What are the measures of ⁀ UXT and ⁀ UYT ?

b. What are the measures of �UTV and �UTW?

c. Do you think that there is a relationship between m ⁀ UXT and m�UTV? Do

you think that there is a relationship between m ⁀ UYT and m�UTW? If so,

what is the relationship?

10. Line TS is tangent to circle Q.

Q

R

S

T

a. Use a straightedge to draw the central angle that is associated with ⁀ RT .

Then use your protractor to measure �RQT and �RTS.

b. How do the measures of the angles compare?

11. Create an Exterior Angles of a Circle Conjecture about the angle measure that

is formed by a tangent and a chord of a circle and the arc measure intercepted

by the chord.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

12. Suppose that a tangent and a secant to a circle intersect, as shown. First, draw

a chord that connects point Q and point T on the circle. Then, use an argument

similar to the one in Question 8 to show that m�QST � 1 __ 2 (m ⁀ QT � m ⁀ RT ).

QR

T

O

U

S

13. Suppose that two tangents to a circle intersect, as shown. First, draw a chord

that connects point B and point D on the circle. Then use an argument similar

to the one in Question 8 to show that m�BCD � 1 __ 2 (m ⁀ BGD � m ⁀ BD ).

A

O

B

C

D

E

G


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

1. An angle with a vertex located in the exterior of a circle can be formed by a

secant and a tangent, two secants, or two tangents.

a. Case 1: Use circle O shown to draw and label an exterior angle formed by a

secant and a tangent.

O

b. Case 2: Use circle O shown to draw and label an exterior angle formed by

two secants.

O

c. Case 3: Use circle O shown to draw and label an exterior angle formed by

two tangents.

O

To prove the Exterior Angles of a Circle Conjecture previously stated, you must prove

each of the three cases.

PROBLEM 3 Proving Conjectures


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

2. Complete the proof for each case.

a. Use the diagram provided to complete the proof for Case 1.

O

T

XRE

A

Given: Secant ‹

___

› EX and tangent

‹

___

› TX intersect at point X.

Prove: m�EXT � 1 __ 2 (m ⁀ ET � m ⁀ RT )

Statements Reasons

1. Secant ‹

___

› EX and tangent

‹

___

› TX intersect at

point X.

1. Given

2. Connect points E and T to form chord ___

ET . Connect points R and T to form

chord ___

RT .

2. Construction


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

b. Use the diagram provided to complete the proof for Case 2.

XA

TE

R

O

Given: Secants ‹

___

› EX and

‹

___

› RX intersect at point X.

Prove: m�EXR � 1 __ 2 (m ⁀ ER � m ⁀ AT )

Statements Reasons

1. Secants ‹

___

› EX and

‹

___

› TX intersect at point X. 1. Given

2. Connect points A and R to form chord ___

AR . 2. Construction


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

c. Use the diagram provided to complete the proof for Case 3.

O

XE

T

AR

Given: Tangents ‹

___

› EX and

‹

___

› AX intersect at point X.

Prove: m�EXT � 1 __ 2 (m ⁀ ERT � m ⁀ ET )

Statements Reasons

1. Tangents ‹

___

› EX and

‹

___

› TX intersect at point X. 1. Given

Congratulations! You have just proven the Exterior Angles of a Circle Conjecture. It is

now known as the Exterior Angles of a Circle Theorem. You can use this theorem as

a valid reason in proofs.

The Exterior Angles of a Circle Theorem states: “If an angle is formed by two

intersecting secants, two intersecting tangents, or an intersecting tangent and

secant of a circle such that the vertex of the angle is in the exterior of the circle,

then the measure of the angle is half of the difference of the measures of the arc(s)

intercepted by the angle.”


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

1. Use the diagrams shown to determine the measures of each.

a. Determine m ⁀ RT .

m ⁀ FG � 86°

m ⁀ HP � 21°

F

R

G

T

P21°

86°

H

W

b. Using the given information, what additional information can you determine

about the diagram?

1

120° 105°2

34

PROBLEM 4 Determine the Measures


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

c. Determine m ⁀ CD .

m ⁀ AB � 88°

m�AED � 80°

B

C

E

A

D

80°88°

d. Explain how knowing m�ERT can help you determine m�EXT.

R

T

EX



11

© 2

010

Car

negi

e Le

arni

ng, I

nc.

Lesson 11.4 | Color Theory 669

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Chords

OBJECTIVESIn this lesson you will:● Determine the relationships between a

chord and a diameter of a circle.● Determine the relationships between

congruent chords and their minor arcs.● Prove the Diameter-Chord Theorem.● Prove the Equidistant Chord Theorem.● Prove the Equidistant Chord Converse

Theorem.● Prove the Congruent Chord-Congruent

Arc Theorem.● Prove the Congruent Chord-Congruent

Arc Converse Theorem.● Prove the Segment-Chord Theorem.

KEY TERMS● Diameter-Chord Theorem● Equidistant Chord Theorem● Equidistant Chord Converse Theorem● Congruent Chord-Congruent Arc Theorem● Congruent Chord-Congruent Arc Converse

Theorem● segments of a chord● Segment-Chord Theorem

11.4 Color Theory

1

Color theory is a set of rules that is used to create color combinations. A color wheel

is a visual representation of color theory. There are many kinds of color wheels;

consider the diagram as an RYB (red-yellow-blue) color wheel.

R: Red

Y: Yellow

B: Blue

P: Purple

O: Orange

G: Green

The color wheel is made of three different kinds of colors: primary, secondary,

and tertiary. Primary colors (red, blue, and yellow) are the colors you start with.

Secondary colors (orange, green, and purple) are created by mixing two primary

colors. Tertiary colors (red-orange, yellow-orange, yellow-green, blue-green, blue-

purple, red-purple) are created by mixing a primary color with a secondary color.

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

1. The locations of the primary colors on the color wheel are points Y (yellow),

R (red), and B (blue) as shown on the circle.

Y

B R

a. Use a straightedge to draw a chord that has endpoints that are primary

colors. What color is created if you mix these two colors?

b. Use your compass and straightedge to draw the perpendicular bisector of

the chord.

c. What do you notice about your perpendicular bisector?

2. Draw a chord ___

AB that does not pass through the center of circle T. Then, use

the following steps to draw a diameter that is perpendicular to chord ___

AB .

T

a. Place your compass point on the center of circle T. Draw an arc that

intersects the chord at two points. Name these point C and point D. Now

open your compass wider than half the distance between ___

CD . Place the

point of the compass on point C and draw an arc toward the center of the

circle. Place the point of the compass on point D and draw an arc toward

the center of the circle. Use your straightedge to draw the diameter that

passes through the intersection of the arcs.

Label the point where the diameter intersects the chord as point P. Label

the point where the diameter intersects ⁀ AB as point Q.

b. How does the length of ___

AP compare to the length of ___

PB ? What does this

tell you about the diameter?

PROBLEM 1 Mixing Primary Colors


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

c. How does the measure of ⁀ AQ compare to the measure of ⁀ BQ ? Explain

your reasoning.

3. Write a Diameter-Chord Conjecture about the effect a circle’s diameter that is

perpendicular to a chord has on that chord and its corresponding arc.

Let’s prove the conjecture.

4. Create a two-column proof of the Diameter-Chord Conjecture.

O

E

D

I

A

M

Given: ___

MI is a diameter of circle O.

___

MI � ___

DA

Prove: ___

MI bisects ___

DA .

___

MI bisects ⁀ DA .

Statements Reasons

1. ___

MI is a diameter of circle O.

___

MI � ___

DA

1. Given

2. Connect points O and D to form

chord ____

OD . Connect points O

and A to form chord ___

OA .

2. Construction


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Congratulations! You have just proven the Diameter-Chord Conjecture. It is now

known as the Diameter-Chord Theorem. You can now use this theorem as a valid

reason in proofs.

The Diameter-Chord Theorem states: “If a circle’s diameter is perpendicular

to a chord, then the diameter bisects the chord and bisects the arc determined by

the chord.”

5. What does ___

TP represent in the relationship between point T and chord ___

AB

in Question 2?

6. Use a straightedge to draw two congruent chords on circle T that are not

parallel to each other. The chords should not be diameters. Label the chords ___

AB and ___

CD .

T

a. For each chord, use your compass and straightedge to draw a line segment

that represents the distance from the center of the circle to the chord. Then

use your compass to compare the lengths of these segments.

b. What do you notice?

7. Write an Equidistant Chord Conjecture about congruent chords and their

distance from the center of the circle.



© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

8. Create a two-column proof of the Equidistant Chord Conjecture.

O

IE

C

H R

D

Given: ___

CH � ___

DR

___

OE � ___

CH

___

OI � ___

DR

Prove: ___

CH and ___

DR are equidistant from the center point.

Hint: You need to get OE � OI.

Statements Reasons

1. ___

CH � ___

DR

___

OE � ___

CH

___

OI � ___

DR

1. Given

2. Connect points O and H, O and

C, O and D, O and R to form radii ____

OH , ____

OC , ____

OD , and ___

OR ,

respectively.

2. Construction

Congratulations! You have just proven the Equidistant Chord Conjecture. It is now

known as the Equidistant Chord Theorem. You can now use this theorem as a valid

reason in proofs.

The Equidistant Chord Theorem states: “If two chords of the same circle or

congruent circles are congruent, then they are equidistant from the center of

the circle.”


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Consider the converse of this theorem.

The Equidistant Chord Converse Theorem states: “If two chords of the same circle

or congruent circles are equidistant from the center of the circle, then the chords are

congruent.”

9. Create a two-column proof of the Equidistant Chord Converse Theorem.

O

IE

C

H R

D

Given: OE � OI ( ___

CH and ___

DR are equidistant from the center point.)

___

OE � ___

CH

___

OI � ___

DR

Prove: ___

CH � ___

DR

Statements Reasons

1. OE � OI

___

OE � ___

CH

___

OI � ___

DR

1. Given


C, O and D, O and R to form

radii ____

OH , ____

OC , ____

OD , and ___

OR ,

respectively.

2. Construction


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

10. Write the Equidistant Chord Theorem and the Equidistant Chord Converse

Theorem as a biconditional statement.

PROBLEM 2 Mixing Primary and Secondary ColorsThe locations of the primary colors on the color wheel are shown on circle T. The

primary colors are points R (red), Y (yellow), and B (blue), and the locations of the

secondary colors are points O (orange), G (green), and P (purple).

T

G

Y

O

R

P

B

1. Use a straightedge to draw two congruent chords. Make sure the chords are

not diameters, and so that one endpoint is a primary color and the other

endpoint is a secondary color. You can use a compass to verify that the chords

are the same length. Write the names of your chords. Identify the colors that

would be created if you mixed the colors of the endpoints of each chord.

2. From each endpoint of each chord, use your straightedge to draw a radius.

Name the central angle formed by each pair of radii. Use a protractor to find

the measures of these central angles. What do you notice?

3. What does Question 2 tell you about the minor arcs formed by the chords?


4. Write a Congruent Chord-Congruent Arc Conjecture about two congruent

chords of a circle and the measures of their corresponding arcs.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

5. Create a two-column proof of the Congruent Chord-Congruent Arc Theorem.

H R

DC

O

Given: ___

CH � ___

DR

Prove: ⁀ CH � ⁀ DR

Statements Reasons

1. ___

CH � ___

DR 1. Given


C, O and D, O and R to form

radii ____

OH , ____

OC , ____

OD , and ___

OR ,

respectively.

2. Construction

Congratulations! You have just proven the Congruent Chord-Congruent Arc

Conjecture. It is now known as the Congruent Chord-Congruent Arc Theorem.

You can now use this theorem as a valid reason in proofs.

The Congruent Chord-Congruent Arc Theorem states: “If two chords of the

same circle or congruent circles are congruent, then their corresponding arcs

are congruent.”

Consider the converse of this theorem.

The Congruent Chord-Congruent Arc Converse Theorem states: “If two arcs

of the same circle or congruent circles are congruent, then their corresponding

chords are congruent.”


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

6. Create a two-column proof of the Congruent Chord-Congruent Arc

Converse Theorem.

H R

DC

O

Given: ⁀ CH � ⁀ DR

Prove: ___

CH � ___

DR

Statements Reasons

1. ⁀ CH � ⁀ DR 1. Given


C, O and D, and O and R to form

radii ____

OH , ____

OC , ____

OD , and ___

OR ,

respectively.

2. Construction

7. Write the Congruent Chord-Congruent Arc Theorem and the Congruent

Chord-Congruent Arc Converse Theorem as a biconditional statement.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Segments of a chord are the segments formed on

O

E

H

R

C

D

a chord when two chords of a circle intersect.

Use the diagram shown to answer Questions 1 through 6.

1. Name the segments of chord ___

HD .

2. Name the segments of chord ___

RC .

3. Use a ruler to measure the length of each segment of chords ___

HD and ___

RC .

4. What do you notice about the product of the lengths of the segments of chord ___

HD and the product of the lengths of the segments of chord ___

RC ?

5. Write a Segment-Chord Conjecture about the products of the lengths of the

segments of two chords intersecting in a circle.

PROBLEM 3 Segments


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

6. Create a two-column proof of the Segment-Chord Conjecture.

Given: Chords HD and RC intersect at point E in circle O.

Prove: EH � ED � ER � EC

Hint: Connect points C and D, and points H and R.

O

E

H

R

C

D

Show the triangles are similar.

Statements Reasons

Congratulations! You have just proven the Segment-Chord Conjecture. It is now

known as the Segment-Chord Theorem. You can use this theorem as a valid reason

in proofs.

The Segment-Chord Theorem states: “If two chords in a circle intersect, then the

product of the lengths of the segments of one chord is equal to the product of the

lengths of the segments of the second chord.”



© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Lesson 11.5 | Solar Eclipses 681

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11OBJECTIVESIn this lesson you will:● Determine the relationship between a

tangent line and a radius.

● Determine the relationship between congruent tangent segments.

● Prove the Tangent Segment Theorem.

● Prove the Secant Segment Theorem.

● Prove the Secant Tangent Theorem.

KEY TERMS● tangent segment● Tangent Segment Theorem● secant segment● external secant segment● Secant Segment Theorem● Secant Tangent Theorem

11.5

Total solar eclipses occur when the moon passes between Earth and the sun. The

position of the moon creates a shadow on the surface of Earth.

A pair of tangent lines forms the boundaries of the umbra, the lighter part of the

shadow. Another pair of tangent lines forms the boundaries of the penumbra, the

darker part of the shadow.

Solar EclipsesTangents and Secants

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 1 Blocking Out the Sun

Consider the tangents shown.

Sun

Moon

Earth

1. For the sun and moon, use a straightedge to draw a radius from the center of

the circle to one of the tangents.

2. Use your protractor to measure the angle between each radius and tangent.

What do you notice?

3. How is the distance from a line to a point not on the line determined?


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

4. Use a compass and straightedge to determine the distance from ‹

___

› AB to point P.

P

A B

5. Draw a circle and label the center. Then draw a line tangent to the circle on the

left or right side of the circle and label the point of tangency. Use a compass

and straightedge to determine the distance from the center point to the point

of tangency.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

6. Explain why the radius of a circle drawn to the point of tangency is always

perpendicular to the tangent line.

7. Can more than one radius of a circle have a perpendicular relationship with a

line drawn tangent to the circle? Explain.

8. What can you conclude about a tangent to a circle and a radius drawn to the

point of tangency?

9. What do you notice about the tangents that form the umbra in the first diagram?

A tangent segment is a segment formed from an exterior point of the circle to the

point of tangency.

10. Choose a point exterior to the circle shown. Label this point P.

O

Use a straightedge to draw a line that passes through point P such that it is tangent

to circle O. Draw a different line that passes through point P such that it is also

tangent to circle O. Label the points of tangency Q and R.

Draw radius ____

OQ and radius ___

OR .

Draw a line segment between P and O.

11. Write a Tangent Segment Conjecture about the tangent segments drawn from

the same point on the exterior of a circle.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11


12. Create a two-column proof of the Tangent Segment Conjecture.

O

T

A

N

Given: ‹

___

› AT is tangent to circle O at point T.

‹

___

› AN is tangent to circle O at point N.

Prove: ___

AT � ___

AN

Statements Reasons

1. ‹

___

› AT is tangent to circle O at point T.

‹

___

› AN is tangent to circle O at point N.

Congratulations! You have just proven the Tangent Segment Conjecture. It is now

known as the Tangent Segment Theorem. You can use this theorem as a valid

reason in proofs.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

The Tangent Segment Theorem states: “If two tangent segments are drawn from

the same point on the exterior of a circle, then the tangent segments are congruent.

13. In the figure, ‹

___

› KP and

‹

___

› KS are tangent to circle W and m�PKS � 46°.

Calculate m�KPS. Explain your reasoning.

W

P

S

K

14. In the figure, ‹

___

› PS is tangent to circle M and m�SMO � 119°.

Calculate m�MPS. Explain your reasoning.

P

S

M

O


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 2 More Segments

You have studied angles located in the exterior of a circle. If the angle was formed

by two secant lines, then each secant line contains a secant segment and an

external secant segment.

A secant segment is formed when two secants intersect in the exterior of a circle.

A secant segment begins at the point at which the two secants intersect, continues

into the circle, and ends at the point at which the secant exits the circle.

An external secant segment is the portion of each secant segment that lies on the

outside of the circle. It begins at the point at which the two secants intersect and

ends at the point where the secant enters the circle.

1. Consider the diagram shown.

C

E

1 1–4

′′

1 1–4

′′

3–8

′′

3–8

′′A

N

S

O

a. Name two secant segments.

b. Name two external secant segments.

c. What do you notice about the product of the lengths of a secant segment

and its external secant segment and the product of the lengths of the

second secant segment and its external secant segment?


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

2. Write a Secant Segment Conjecture about the products of the lengths of a

secant segment and its external secant segment of two secants intersecting

outside a circle.

3. Create a two-column proof of the Secant Segment Conjecture.

Given: Secants ‹

___

› CS and

‹

___

› CN intersect at point C in the exterior of circle O.

Prove: CS � CE � CN � CA

Hint: Connect points A and S, and points E and N.

O

Statements Reasons

Congratulations! You have just proven the Secant Segment Conjecture. It is now known

as the Secant Segment Theorem. You can use this theorem as a valid reason in proofs.

The Secant Segment Theorem states: “If two secants intersect in the exterior of a

circle, then the product of the lengths of the secant segment and its external secant

segment is equal to the product of the lengths of the second secant segment and its

external secant segment.”


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

4. Consider an angle formed in the exterior of a circle by a secant and a tangent

shown in the diagram.

T

N

A

G

O

3 cm

1.5 cm

4 cm

a. Name a tangent segment.

b. Name a secant segment and an external secant segment.

c. What do you notice about the product of the lengths of a secant segment

and its external secant segment and the square of the length of the

tangent segment?

5. Write a Secant Tangent Conjecture about the product of the lengths of a secant

segment and its external secant segment and the square of the length of the

tangent segment.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

6. Create a two-column proof of the Secant Tangent Conjecture.

Given: Tangent ‹

___ › AT and secant

‹

___

› AG intersect at point A in the exterior of circle O.

Prove: (AT)2 � AG � AN

Hint: Connect points N and T, and points G and T.

O

Statements Reasons

Congratulations! You have just proven the Secant Tangent Conjecture. It is now known

as the Secant Tangent Theorem. You can use this theorem as a valid reason in proofs.

The Secant Tangent Theorem states: “If a tangent and a secant intersect in the

exterior of a circle, then the product of the lengths of the secant segment and its

external secant segment is equal to the square of the length of the tangent segment.”


Lesson 11.6 | Replacement for a Carpenter’s Square 691

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11Inscribed and Circumscribed Triangles and Quadrilaterals

OBJECTIVESIn this lesson you will:● Determine a property of a triangle

inscribed in a circle.● Determine how to circumscribe

a triangle about a circle.● Determine a property of a quadrilateral

inscribed in a circle.● Determine how to circumscribe a

quadrilateral about a circle.● Prove the Inscribed Right Triangle-

Diameter Theorem.● Prove the Inscribed Right Triangle-

Diameter Converse Theorem.● Prove the Inscribed Quadrilateral-

Opposite Angles Theorem.

KEY TERMS● inscribed polygon● Inscribed Right Triangle-Diameter Theorem● Inscribed Right Triangle-Diameter

Converse Theorem● circumscribed polygon● Inscribed Quadrilateral-Opposite

Angles Theorem

11.6 Replacement for aCarpenter’s Square

A carpenter’s square is a tool that is used to create right angles. These “squares”

are usually made of a strong material like metal so that the right angle is not easily

bent or broken.

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

A carpenter is working on building a children’s playhouse. She accidentally drops

her carpenter’s square and the right angle gets bent. She still needs to cut out a

piece of plywood that is in the shape of a right triangle. So the carpenter gets out her

compass and straightedge to get the job done.

1. Use the steps to recreate how the carpenter created the right triangle.

a. The hypotenuse of the triangle needs to be 6 centimeters. Use your ruler

and open your compass to 3 centimeters. In the space provided, draw a

circle with a diameter of 6 centimeters. Use the given point as the center.

b. Use your straightedge to draw a diameter on the circle.

c. One of the legs of the triangle is to be 4 centimeters long. Open your

compass to 4 centimeters. Place the point of your compass on one of the

endpoints of the diameter and draw an arc that passes through the circle.

d. Use your straightedge to draw segments from the endpoints of the diameter

to the intersection of the circle and the arc.

e. Use your protractor to verify that this triangle is a right triangle.

An inscribed polygon is a polygon drawn inside a circle such that each vertex of

the polygon touches the circle.

PROBLEM 1 In Need of a New Tool


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

2. Consider � ABC that is inscribed in circle P.

A

B

C

D

P

a. What do you know about ___

AC ?

b. What do you know about m ⁀ ADC ? Explain your reasoning.

c. What does this tell you about m�ABC? Explain your reasoning.

d. What kind of triangle is �ABC? How do you know?

3. Write an Inscribed Right Triangle-Diameter Conjecture about the kind of

triangle inscribed in a circle when one side of the triangle is a diameter.

4. Write the converse of the conjecture you wrote in Question 3. Do you

think this statement is also true?


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

5. Use your compass to draw a circle. Then use your protractor and straightedge

to draw an inscribed angle in the circle so that the angle is a right angle. Finally,

use your straightedge to complete the triangle.

a. Is one of the sides a diameter? Which side?

b. Consider the intercepted arc of your right angle. What is its measure? Why

does this tell you that one of the sides of the triangle must be a diameter?


Let’s prove the conjectures!

6. Create a two-column proof of the Inscribed Right Triangle-Diameter Conjecture.

Given: �HYP is inscribed in circle O such that ___

HP is the diameter of the circle.

Prove: �HYP is a right triangle.

Statements Reasons


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Congratulations! You have just proven the Inscribed Right Triangle-Diameter

Conjecture. It is now known as the Inscribed Right Triangle-Diameter Theorem.

You can use this theorem as a valid reason in proofs.

The Inscribed Right Triangle-Diameter Theorem states: “If a triangle is inscribed

in a circle such that one side of the triangle is a diameter of the circle, then the

triangle is a right triangle.”

7. Create a two-column proof of the Inscribed Right Triangle-Diameter

Converse Conjecture.

OH P

R

Y

Given: Right �HYP is inscribed in circle O.

Prove: ___

HP is the diameter of circle O.

Statements Reasons

Congratulations! You have just proven the Inscribed Right Triangle-Diameter

Converse Conjecture. It is now known as the Inscribed Right Triangle-Diameter

Converse Theorem. You can use this theorem as a valid reason in proofs.

The Inscribed Right Triangle-Diameter Converse Theorem states: “If a right

triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.”

A circumscribed polygon is a polygon drawn outside a circle such that each

side of the polygon is tangent to a circle.

Mr. Scalene asked his geometry class to draw a triangle and use a compass

to draw a circle inside the triangle such that the circle was tangent to each side

of the triangle. See if you perform this task.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

8. Use a straightedge to draw a triangle.

a. Use your compass to draw a circle inside the triangle such that each side

of the triangle is tangent to the circle.

b. Were you able to do it? Explain.

Rachel in Mr. Scalene’s class did this task with no problem. Mr. Scalene asked

Rachel to give others in the class a hint. She smiled and said, “It has something

to do with what we learned about a point of concurrency!”

c. Use your triangle to discover how the student was able to easily draw

a circumscribed triangle and explain how you were able to do it.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

1. Use your compass to draw a circle.

a. Use your straightedge to draw an inscribed quadrilateral that is not a

parallelogram in your circle. Label the vertices of your quadrilateral.

b. Use your protractor to find the measures of the angles of the quadrilateral.

What is the relationship between the measures of each pair of

opposite angles?

c. Write an Inscribed Quadrilateral-Opposite Angles Conjecture about the

opposite angles of an inscribed quadrilateral.

Let’s prove the conjecture!

PROBLEM 2 Quadrilaterals and Circles


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

2. Create a two-column proof of the Inscribed Quadrilateral-Opposite Angles

Conjecture.

O A

D

Q

U

Given: Quadrilateral QUAD is inscribed in circle O.

Prove: �Q and �A are supplementary angles.

�U and �D are supplementary angles.

Statements Reasons

Congratulations! You have just proven the Inscribed Quadrilateral-Opposite Angles

Conjecture. It is now known as the Inscribed Quadrilateral-Opposite Angles Theorem.

You can use this theorem as a valid reason in proofs.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

The Inscribed Quadrilateral-Opposite Angles Theorem states: “If a quadrilateral is

inscribed in a circle, then the opposite angles are supplementary.”

Mrs. Rhombi asked her geometry class to draw a quadrilateral and use a compass

to draw a circle inside the quadrilateral such that the circle was tangent to each side

of the quadrilateral. See if you can do it.

Most of her students used a straightedge and compass and drew this quadrilateral

and tried to draw a circle inside the quadrilateral as shown.

One of her students talked to a friend who happened to be in Mr. Scalene’s class

so she knew exactly what to do if it was a triangle. Will this information help her if it

is a quadrilateral?

Are you able to circumscribe the quadrilateral the same way you circumscribed

the triangle?

3. Use the quadrilateral shown and try it for yourself. Explain your process.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

4. Ms. Rhombi then wrote the following theorem on the blackboard:

A circle can be inscribed in a quadrilateral if and only if the angle

bisectors of the four angles of the quadrilateral are concurrent.

Using this theorem, how can you tell if it is possible to inscribe a circle

in the quadrilateral in Question 3?

Is it possible to inscribe a circle in this quadrilateral?


Lesson 11.7 | Gears 701

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11OBJECTIVEIn this lesson you will:● Determine the length of an arc

of a circle.

KEY TERM● arc length

11.7

Gears are used in many mechanical devices to provide torque, or the force that

causes rotation. For instance, an electric screwdriver contains gears. The motor of an

electric screwdriver can make the spinning components spin very fast, but the gears

are needed to provide the force to push a screw into place. Gears can be very large

or very small, depending on their application. Often gears work together, such as the

gears shown.

GearsArc Length

PROBLEM 1 Large and Small Gears

Consider the circles shown that model two gears that work together.

A

B

1. Use your protractor to draw a central angle on each circle that has

a measure of 60º.

2. What is the measure of each of the minor arcs associated with these

central angles?

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

3. Write a fraction that compares the degree measure of the minor arc to the

degree measure of the entire circle.

4. When gears are used together, the circumference of the gears are important

because the gears move together. Describe the circumference of a circle.

5. Which gear in Problem 1 has a greater circumference? Explain your reasoning.

6. What fraction of the circumference do you think is taken up by the minor arcs

described previously? Explain your reasoning.

A portion of the circumference of a circle is called an arc length.

7. Suppose that the circumference of circle A is 48� inches and the circumference

of circle B is 36� inches. What are the lengths of the minor arcs? Express your

answer in terms of �.

8. Consider the minor arcs of the central angles you drew. How do the measures

of the arcs compare? How do the lengths of the arcs compare?

9. How is the measure of an arc different from the length of an arcs? Express your

answer in terms of �.

10. Use complete sentences to explain how you can calculate the length of an arc

when you know its measure.

a. Consider the circle shown. What is the circumference of the circle?

Express your answers in terms of �.

r = 4 meters


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

b. Choose two points on the circle and label the points as point A and

point B. Then, use a protractor to determine the measure of ⁀ AB . Arc AB

is what fraction of the circle?

c. Use your answers to write an expression for the arc length of ⁀ AB .

Express your answers in terms of �.

11. Write an expression to represent the length of ⁀ AB in the circle shown.

A

Br

12. Calculate the arc length of each circle. Express your answer in terms of �.

a. 10 inches

A

B80�

b. 10 inches

A

B

120�

c.

20 inches

A

B80�

d.

20 inches

A

B

120�


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

13. Describe the relationship between the measure of an arc and its arc length.

14. Two semicircular cuts were taken from the rectangular region shown.

Determine the perimeter of the shaded region. Do not express your answer in

terms of �.

8

20


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

15. Use the diagram shown to answer each question.

a. The radius of a small tree ring (small circle) is r and the radius of a larger tree

ring (large circle) is 10r. How does the arc length of the minor arc in the small

tree ring compare to the arc length of the minor arc in the large tree ring?

b. If the arc length of the minor arc in the small tree ring is equal to 3 inches,

what is the arc length of the minor arc in the large tree ring?

c. If m�A � 20°, the length of the radius of the small tree ring is r, the length

of the radius of the large tree ring is 10r, and the length of the minor arc of

the small tree ring is 3 inches, determine the circumference of the large

tree ring.

d. Did you have to know the length of the radius to determine

the circumference?


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

16. Joan wanted to measure the circumference of the tree in her front yard.

She wrapped a piece of rope around a tree trunk and measured the length

of the rope. If the length of the rope was 4.5 feet, determine the length of the

radius of the tree. Do not express your answer in terms of �.


Lesson 11.8 | Playing Darts 707

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Sectors and Segments of a Circle

OBJECTIVESIn this lesson you will:● Determine the area of sectors

of a circle.● Determine the area of segments

of a circle.

KEY TERMS● sector of a circle● segment of a circle

11.8 Playing Darts

Consider the following diagram of a standard dartboard. Each different section of

the board is surrounded by wire and the numbers indicate scoring for a game. There

are different games with different scoring that can be played on a dartboard, but the

highest score from a single throw occurs when a dart lands at the very center, or

bullseye, of the dartboard.

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

The dartboard is made of concentric circles which are two circles that have the same

center, but different radii.

1. The first circle inside the outermost circle of the dartboard has a diameter of 170

millimeters. Calculate the area of this circle. Express your answer in terms of �.

2. Imagine that the pie-shaped sections extend to the center of the circle. How

many pie-shaped sections is this circle divided into?

3. What is the measure of the central angle formed by one of these pie-shaped

sections if all of the sections are congruent? Explain your reasoning.

4. What is the measure of the minor arc associated with this central angle?

5. What fraction of the circle is the minor arc?

6. What fraction of the circle’s area is covered by one of the pie-shaped sections?


7. How do the fractions in Questions 6 and 7 compare?

8. What is the area of one pie-shaped section? Express your answer in terms of �.

PROBLEM 1 Hitting the Bullseye


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

A sector of a circle is a region of the circle bounded by two radii and the

included arc.

9. How can the sides of the pie-shaped section be described with respect

to the circle?

10. Draw and shade a sector on the circle shown. Label the points that form

the sector. Name the radii and the included arc that define the sector.

O

11. Describe how to determine the area of a sector if the length of the radius and

the measure of the included arc are known.

12. Use the figure shown to write an expression representing the area of the sector.

A

BrO

13. Use the figure to write an expression representing the arc length of the

included arc.

14. Compare the two expressions. What is the same? What is different?

Consider the dartboard in Problem 1. Suppose the innermost circle divided into

20 sectors and has a diameter of 108 millimeters. Notice that half of the sectors on

the dartboard are the same color.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

15. Describe two methods for calculating the total area of all sectors of the same

color. Then calculate this area and the area of one sector. Express your

answers in terms of �.

A segment of a circle is a region of the circle bounded by a chord and the

included arc.

A

C B

16. Name the chord and arc that bound the shaded segment of the circle.

17. Describe a method for calculating the area of the segment of the circle.

18. If the length of the radius of circle C is 8 centimeters and m�ACB � 90°,

use your method to determine the area of the shaded segment of the circle.

Express your answer in terms of �. Then rewrite your answer rounded to the

nearest hundredth.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

19. The area of the segment shown is 9� � 18 square feet. Calculate the radius

of circle O.

OB

A

20. The area of the segment is 10.26 square feet. Calculate the radius of circle O.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

21. The length of the radius is 10 inches. Calculate the area of the shaded region

of circle O. Express your answer in terms of �.

OB

A


Lesson 11.9 | The Coordinate Plane 713

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11OBJECTIVESIn this lesson you will:● Apply theorems to circles in a coordinate plane.

● Classify polygons in the coordinate plane.

● Use midpoints to determine characteristics of polygons.

● Distinguish between showing something is true under certain conditions, and proving it is always true.

11.9

This lesson provides you an opportunity to review several familiar theorems and

classify polygons formed in a coordinate plane. Circles and polygons located in a

coordinate plane enable you to calculate distances, slopes, and equations of lines.

It is important to understand the difference between showing something is true under

certain conditions and proving something is always true. You will experience and

differentiate between both instances in the following problems.

The Coordinate PlaneCircles and Polygons on the Coordinate Plane

11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 1

OP � 2.5 units

x86

Y

OH P

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

HY � 4 units

PY � 3 units

The figure shown is circle O with the

center point at the origin. The length of

the radius is 2.5 units.

1. Determine m ___

HP .

2. Use a compass and the grid shown to construct:

● all points 4 units from (�2.5, 0)

● all points 3 units from (2.5, 0)

x86

Y

O

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

P (2.5, 0)H (–2.5, 0)

3. Label the point at which both constructed circles intersect point Y.

4. Is �HYP a right triangle? Justify your conclusion.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

5. Where is point Y located with respect to circle O?

6. What kind of angle is �HYP with respect to circle O?

7. Describe the arc intercepted by �HYP.

8. Describe the chord determined by the intercepted arc.

9. Which side is the hypotenuse of the right triangle?

10. Is the hypotenuse of the right triangle also a diameter of the circle?

11. What theorem does this problem illustrate?

12. Is this an instance of showing something is true under certain conditions,

or proving something is always true?


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 2

Line ‹

___

› AT and

‹

___

› AN intersect at point A and are drawn tangent to circle O at points T

and N, respectively. The coordinates of point T are (4, 3) and the coordinates of

point N are (0, �5). The center of circle C is at the origin and the length of the

radius is 5.

x86

T (4, 3)

O

C (0, 0)

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

N (0, –5)

A

1. To determine if the length of the tangent segments are equal, what

additional information is needed?

2. Describe a strategy to determine the additional information needed.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

3. Determine the slope of each radius.

4. Determine the slope of each tangent line.

5. Determine an equation for each tangent line.

6. Determine the coordinates of point A.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

7. Show the lengths of the tangent segments are equal.




PROBLEM 3

Line ‹

___

› AT is tangent to circle C at point T. Secant

‹

___

› AG intersects circle C at

points N and G. Line ‹

___

› AT intersects

‹

___

› AG at point A. The coordinates of point A are (10, 0).

The center of circle C is at the origin and the length of the radius is 6.

x86

T

N

O

G

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

A (10, 0)C (0, 0)

1. Describe a strategy to show the product of the secant segment and the external

secant segment is equal to the square of the length of the tangent segment.


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 4

1. Is the quadrilateral formed by connecting the midpoints of the sides of a square

also a square? The coordinates on the diagram shown define a square so it is

important to remember x � y.

x

y

(0, y) (x, y)

(x, 0)(0, 0)




© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 5

1. Draw four points on the circle. Provide the coordinates and labels for each

point. Connect the points to form a quadrilateral. Classify the polygon formed

by connecting the midpoints of the sides of the quadrilateral.

x86O

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

2. Is this an instance of showing something is true under certain conditions, or

proving something is always true?


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

PROBLEM 6

1. Is a rhombus formed by connecting the midpoints of the sides of an

isosceles trapezoid?

a. Draw an isosceles trapezoid.

b. Inscribe the trapezoid in a circle.

c. Label the vertices of the trapezoid.

d. Choose reasonable coordinates for each vertex.

e. Use the coordinates to determine if this conjecture could be true.

x43

N

1

3

4

–1–1

21–2

–2

–3

–3

–4

–4

y

2


© 2

010

Car

negi

e Le

arni

ng, I

nc.

PROBLEM 7 Summary

1. Describe the difference between showing something is true under certain

conditions and proving something to be true under all conditions.




11


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Chapter 11 Checklist

KEY TERMS● center of a circle (11.1)● chord (11.1)● secant of a circle (11.1)● tangent of a circle (11.1)● point of tangency (11.1)● central angle (11.1)● inscribed angle (11.1)● arc (11.1)● major arc (11.1)

● minor arc (11.1)● semicircle (11.1)● degree measure

(of an arc) (11.2)● adjacent arcs (11.2)● intercepted arc (11.2)● segments of a chord (11.4)● tangent segment (11.5)● secant segment (11.5)

● external secant segment (11.5)

● inscribed polygon (11.6)● circumscribed polygon (11.6)● arc length (11.7)● sector of a circle (11.8)● segment of a circle (11.8)

POSTULATE● Arc Addition Postulate (11.2)

THEOREMS● Inscribed Angle

Theorem (11.2)● Parallel Lines-Congruent

Arcs Theorem (11.2)● Interior Angles of a Circle

Theorem (11.3)● Exterior Angles of a Circle

Theorem (11.3)● Diameter-Chord

Theorem (11.4)● Equidistant Chord

Theorem (11.4)

● Equidistant Chord Converse Theorem (11.4)

● Congruent Chord-Congruent Arc Theorem (11.4)

● Congruent Chord-Congruent Arc Converse Theorem (11.4)

● Segment-Chord Theorem (11.4)

● Tangent Segment Theorem (11.5)

● Secant Segment Theorem (11.5)

● Secant Tangent Theorem (11.5)

● Inscribed Right Triangle-Diameter Theorem (11.6)

● Inscribed Right Triangle-Diameter Converse Theorem (11.6)

● Inscribed Quadrilateral-Opposite Angles Theorem (11.6)

CONSTRUCTION● circumscribed polygons (11.6) ©

201

0 C

arne

gie

Lear

ning

, Inc

.

11

Chapter 11 | Checklist 725

© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Identifying Parts of a Circle

A circle is a locus of points on a plane equidistant from a given point. The following

are parts of a circle.

• The center of a circle is a point inside the circle that is equidistant from every

point on the circle.

• A radius of a circle is a line segment that is the distance from a point on the

circle to the center of the circle.

• A chord is a segment whose endpoints are on a circle.

• A diameter of a circle is a chord across a circle that passes through the center.

• A secant is a line that intersects a circle at exactly two points.

• A tangent is a line that intersects a circle at exactly one point, and this point is

called the point of tangency.

• A central angle is an angle of a circle whose vertex is the center of the circle.

• An inscribed angle is an angle of a circle whose vertex is on the circle.

• A major arc of a circle is the largest arc formed by a secant and a circle.

• A minor arc of a circle is the smallest arc formed by a secant and a circle.

• A semicircle is exactly half a circle.

Examples:

Point A is the center of circle A.

A

B

C

D

E

F

G

Segments AB, AC, and AE are radii of circle A.

Segment BC is a diameter of circle A.

Segments BC, DC, and DE are chords of circle A.

Line DE is a secant of circle A.

Line FG is a tangent of circle A, and point C is a

point of tangency.

Angle BAE and angle CAE are central angles.

Angle BCD and angle CDE are inscribed angles.

Arcs BDE, CDE, CED, DCE, and DCB are major arcs.

Arcs BD, BE, CD, CE, and DE are minor arcs.

Arc BDC and arc BEC are semicircles.

11.1


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Determining Measures of Arcs

The degree measure of a minor arc is the same as the degree measure of its

central angle.

Example:

In circle Z, �XZY is a central angle measuring 120°. So, m ⁀ XY � 120°.

X

YZ120°

Using the Arc Addition Postulate

Adjacent arcs are two arcs of the same circle sharing a common endpoint. The Arc

Addition Postulate states: “The measure of an arc formed by two adjacent arcs is

equal to the sum of the measures of the two arcs.”

Example:

In circle A, arcs BC and CD are adjacent arcs. So, m ⁀ BCD � m ⁀ BC � m ⁀ CD �

180° � 35° � 215°.

B

D

A

35°

C

11.2

11.2


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Using the Inscribed Angle Theorem

The Inscribed Angle Theorem states: “The measure of an inscribed angle is one half

the measure of its intercepted arc.”

Example:

In circle M, �JKL is an inscribed angle whose intercepted arc JL measures 66°.

So, m�JKL � 1 __ 2 (m ⁀ JL ) � 1 __

2 (66°) � 33°.

L

M

66°

J

K

Using the Parallel Lines-Congruent Arcs Theorem

The Parallel Lines-Congruent Arcs Theorem states: “Parallel lines intercept congruent

arcs on a circle.”

Example:

Lines AB and CD are parallel lines on circle Q and m ⁀ AC � 60°. So, m ⁀ AC � m ⁀ BD ,

and m ⁀ BD � 60°.

D

QC

A

m�AC = 60°B

11.2

11.2


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Using the Interior Angles of a Circle Theorem

The Interior Angles of a Circle Theorem states: “If an angle is formed by two

intersecting chords or secants such that the vertex of the angle is in the interior of

the circle, then the measure of the angle is half the sum of the measures of the arcs

intercepted by the angle and its vertical angle.”

Example:

In circle P, chords QR and ST intersect to form vertex angle TVR and its vertical

angle QVS. So, m�TVR � 1 __

2 (m ⁀ TR � m ⁀ QS ) � 1 __

2 (110° � 38°) � 1 __

2 (148°) � 74°.

T

S

P

R

Q

V

m�TR = 110°

m�QS = 38°

Using the Exterior Angles of a Circle Theorem

The Exterior Angles of a Circle Theorem states: “If an angle is formed by two

intersecting secants, two intersecting tangents, or an intersecting tangent and secant

such that the vertex of the angle is in the exterior of the circle, then the measure of

the angle is half the difference of the measures of the arc(s) intercepted by

the angle.”

Example:

In circle C, secant FH and tangent FG intersect to form vertex angle GFH.

So, m�GFH � 1 __ 2 (m ⁀ GH � m ⁀ JG ) � 1 __

2 (148° � 45°) � 1 __

2 (103°) � 51.5°.

H

C

G

F

Jm�JG = 45°

m�GH = 148°

11.3

11.3


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Using the Diameter-Chord Theorem

The Diameter-Chord Theorem states: “If a circle’s diameter is perpendicular to

a chord, then the diameter bisects the chord and bisects the arc determined by

the chord.”

Example:

In circle K, diameter ___

ST is perpendicular to chord ___

FG . So FR � GR and m ⁀ FT � m ⁀ GT .

SK

G

F

TR

Using the Equidistant Chord Theorem and the Equidistant Chord Converse Theorem

The Equidistant Chord Theorem states: “If two chords of the same circle or

congruent circles are congruent, then they are equidistant from the center

of the circle.”

The Equidistant Chord Converse Theorem states: “If two chords of the same circle

or congruent circles are equidistant from the center of the circle, then the chords

are congruent.”

Example:

In circle A, chord ___

CD is congruent to chord ___

XY . So PA � QA.

C

P

Y

D

Q

AX

11.4

11.4


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Using the Congruent Chord-Congruent Arc Theorem and the Congruent Chord-Congruent Arc Converse Theorem

The Congruent Chord-Congruent Arc Theorem states: “If two chords of the same

circle or congruent circles are congruent, then their corresponding arcs

are congruent.”

The Congruent Chord-Congruent Arc Converse Theorem states: “If two arcs of the

same circle or congruent circles are congruent, then their corresponding chords

are congruent.

Example:

In circle X, chord ___

JK is congruent to chord ___

QR . So m ⁀ JK � m ⁀ QR .

J

Q

K

X

R

Using the Segment-Chord Theorem

Segments of a chord are the segments formed on a chord when two chords of a

circle intersect.

The Segment-Chord Theorem states: “If two chords in a circle intersect, then the

product of the lengths of the segments of one chord is equal to the product of the

lengths of the segments of the second chord.”

Example:

In circle H, chords ___

LM and ____

VW intersect to form ___

LK and ____

MK of chord ___

LM and ____

WK

and ___

VK of chord ____

VW . So LK � MK � WK � VK.

LM

W

H

V

K

11.4

11.4


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Using the Tangent Segment Theorem

A tangent segment is a segment formed from an exterior point of a circle to the point

of tangency.

The Tangent Segment Theorem states: “If two tangent segments are drawn from the

same point on the exterior of a circle, then the tangent segments are congruent.”

Example:

In circle Z, tangent segments ___

SR and ___

ST are both drawn from point S outside the

circle. So, SR � ST.

S T

R

Z

Using the Secant Segment Theorem

A secant segment is a segment formed when two secants intersect in the exterior of

a circle. An external secant segment is the portion of a secant segment that lies on

the outside of the circle.

The Secant Segment Theorem states: “If two secant segments intersect in the

exterior of a circle, then the product of the lengths of the secant segment and its

external secant segment is equal to the product of the lengths of the second secant

segment and its external secant segment.”

Example:

In circle B, secant segments ____

GH and ___

NP intersect at point C outside the circle.

So, GC � HC � NC � PC.

GH

N

B P

C

11.5

11.5


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Using the Secant Tangent Theorem

The Secant Tangent Theorem states: “If a tangent and a secant intersect in the

exterior of a circle, then the product of the lengths of the secant segment and

its external secant segment is equal to the square of the length of the

tangent segment.”

Example:

In circle F, tangent ___

QR and secant ___

YZ intersect at point Q outside the circle.

So, QY � QZ � QR 2.

R

Z

Y

F

Q

Using the Inscribed Right Triangle-Diameter Theorem and the Inscribed Right Triangle-Diameter Theorem

The Inscribed Right Triangle-Diameter Theorem states: ”If a triangle is inscribed in a

circle such that one side of the triangle is a diameter of the circle, then the triangle is

a right triangle.”

The Inscribed Right Triangle-Diameter Converse Theorem states: ”If a right triangle is

inscribed in a circle, then the hypotenuse is a diameter of the circle.”

Examples:

Triangle RST is inscribed in circle Q. Because ___

RT of �RST

R

S

TQ

is a diameter of circle Q, �RST is a right triangle.

Triangle ABC is inscribed in circle D. Because

�ABC is a right triangle, the hypotenuse of

�ABC, ___

BC , is a diameter of circle D.

A

B

C

D

11.5

11.6


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Using the Inscribed Quadrilateral-Opposite Angles Theorem

The Inscribed Quadrilateral-Opposite Angles Theorem states: ”If a quadrilateral is

inscribed in a circle, then opposite angles are supplementary.”

Example:

L

M

N

J

K

Because quadrilateral KLMN is inscribed in circle J, the opposite angles of

quadrilateral KLMN are supplementary. So, �K and �M are supplementary and, �L

and �N are supplementary.

Calculating Arc Lengths

To calculate the arc length of a circle, divide the measure of the arc by 360°, then

multiply the result by the circumference of the circle.

Example:

The arc length of ⁀ GH is 60° _____ 360°

� 2� (14) � 1 __ 6 � 28� � 14 ___

3 � inches.

H

FG60°

14 in.

11.6

11.7


© 2

010

Car

negi

e Le

arni

ng, I

nc.

11

Determining the Area of a Sector of a Circle

A sector of a circle is a region of the circle bounded by two radii and the included

arc. To calculate the area of a sector of a circle, divide the measure of the included

arc by 360°, then multiply the result by the area of the circle.

Example:

The area of sector BAC is 160° _____ 360°

� �(92) � 4 __

9 � 81� � 36� square feet.

C

A

B

160°

9 ft

Determining the Area of a Segment of a Circle

A segment of a circle is a region of the circle bounded by a chord and the included

arc. To calculate the area of a segment of a circle, subtract the area of the triangle

formed by the chord and the radii from the area of the sector.

Example:

The area of sector YXZ is 90° _____ 360°

� �(302) � 1 __ 4 � 900� � 225� square yards. The area of

triangle XYZ is 1 __ 2 (30)(30) � 450 square yards. So, the area of the segment of circle X

formed by ___

YZ and ⁀ YZ is 225� � 450 � 256.9 square yards.

Z

X

Y

30 yds

Proving Characteristics of Circles and Polygons in the Coordinate Plane

To prove that a statement involving circles or polygons in the coordinate plane is true

under certain conditions, express the hypothesis and conclusion using appropriate

values. To prove that a statement involving circles or polygons in the coordinate

plane is true under all conditions, express the hypothesis and conclusion

using variables.

11.8

11.8

11.9

© 2010 carnegie learning, inc. - oak park geometry classes€¦ · 11 chapter 11 | circles 633...

Documents