geometry (grades 9-10)geometry (grades 9-10) 1 charles county public schools geometry (grades 9-10)...

Geometry (Grades 9-10)

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CHARLES COUNTY PUBLIC SCHOOLS

Geometry (Grades 9-10) Mathematics

Weeks 7-8 (May 18 – May 29)

Dear parents,

If your child is participating in distance learning solely through the completion of our instructional packets, you are required to call or email the principal to inform them of your child’s participation status, since packet-assignments will not be collected until a later time. Please keep all of your child’s work in a safe place until you are notified of when, where and how to submit. Thank you for your attention to this matter.


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Estimados padres, Si su hijo/a está participando en el aprendizaje a distancia completando solamente nuestros paquetes de instrucción, deberá llamar o enviar un correo electrónico al director para informarle sobre el estado de participación de su hijo/a, ya que las asignaciones realizadas en los paquetes no se recopilarán hasta más tarde. Por favor mantenga todo el trabajo de su hijo/a en un lugar seguro hasta que se le notifique cuándo, dónde y cómo presentarlo. Gracias por su atención a este asunto.


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Student: _________________________________ School: _____________________________

Teacher: _________________________________ Block/Period: ________________________

Packet Directions for Students

Week 7:

Read through the Instruction and examples on Chords while completing the corresponding questions on the 9.2.1 Study: Chords Study Guide.

Complete 9.2.1 Study: Chords Study Guide o Check and revise solutions using the 9.2.1 Study: Chords Study Guide Answer Key

Complete Quiz: Congruent Chords

Week 8:

Read through the Instruction and examples on Arcs while completing the corresponding questions on the 9.3.1 Study: Arcs study guide.

Complete 9.3.1 Study: Arcs study guide. o Check and revise solutions using the 9.3.1 Study: Arcs study guide Answer Key

Complete Quiz: Arc Types and Measure


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Chords

There are a number of important and interesting relationships between the parts of a circle.

There is the center and the radii and a part called a chord. A special chord is called the

diameter.

You will learn more about the parts of a circle in this lesson. Take a look at the objectives listed

below before you begin.

Objectives

Define diameter and identify the diameter of a circle.

Define chord and identify chords in a circle.

Discover and apply the properties of congruent chords.

Prove that a radius bisects a chord if it is perpendicular to the chord.

Determine the properties of a perpendicular bisector of a chord.


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Chords

This lesson will introduce you to two special types of segments on a circle — the chord and

the diameter.

Before you learn these, make sure you are comfortable with the three basic parts of circles

you've seen so far. Match each part with its definition below.

Select each item in the left column and its match in the right column.

Definition of a Chord

The first special segment on a circle is a chord.


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Distance from the Center to a Chord

A chord is a line segment that has both endpoints on the circle.

Throughout this unit, you'll be solving problems involving circles and chords. Sometimes,

knowing the distance between a chord and the center of the circle will be useful in solving

those problems.

Using Perpendicular Bisectors to Find Distance

You may recall that the shortest distance between a line segment and a point not on the

segment is the length of the connecting perpendicular.

Examples:


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What If Perpendicular Bisectors Have Equal Lengths?

You probably saw that the length of the perpendicular bisector increased as you moved

the chord farther away from the center of the circle. Similarly, the length decreased as the

chord moved closer to the center.

But what if you are working with two chords whose perpendicular bisectors both have

the same length? Does that mean anything?

Something special happens when two chords are the same distance from the center of the

circle. Use the tool below to see if you can figure out what happens.

Congruent Chords

You might have noticed that chords that are the same distance from the center of the

circle have the same length. In fact, chords that have the same length are congruent.

The Converse — True or False?

You just saw that two chords that are the same distance from the center of the

circle are congruent.

Is the converse true? If two chords are congruent, are they the same distance from the center

of a circle?

Yes! If two chords are congruent they are the same distance from the center.


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Confirm — Congruent Chords

You have seen that segments with both endpoints on a circle are called chords.

You also learned the congruent chord rule and its converse.

A Radius Perpendicular to a Chord

You have seen how to use congruent angles to find out more about a circle. Now take a look at another

special relationship between circles and chords.

If you noticed that a radius that is perpendicular to the chord divides the chord into two equal pieces,

you are right!


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The Converse — True or False?

Now it's time to test the converse. You just proved that if a radius of a circle is perpendicular to a chord,

then it bisects that chord. Is the converse true? That is, if a chord is bisected by a radius, is the radius

perpendicular to the chord?

The Converse Is True

It turns out that the converse is true again! A radius is always perpendicular to the chord it bisects.

Confirm — Chords and Perpendicular Radii

Over the previous pages, you proved that both the perpendicular radii rule and its converse are true.

Discovering Diameter

You are now ready to learn about the second type of special segment on a circle.

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


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The Diameter of a Circle

You may have noticed that a diameter is the same as two radii that extend from the center of the

circle in opposite directions. So the length of the diameter must be two times the radius of a circle.

You learned earlier that all radii in a given circle have the same length. That means all diameters must

have the same length as well. As with a radius, the word "diameter" can mean both the line segment

and its length.

Confirm — Diameter

Take a minute to confirm that you know the formal definition of this term.


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9.2.1 Study: Chords Study Guide

Name:

Date:

Use the questions below to keep track of key concepts from this lesson's study activity.

1) Practice: Summarizing

Define each term in 15 words or less.

chord: _________________________________________

_______________________________________________

_______________________________________________

_______________________________________________

diameter: ______________________________________

_______________________________________________

_______________________________________________

_______________________________________________

2) Practice: Using Visual Cues and Summarizing

Draw three chords and three diameters on the circles below. Label their endpoints.

Use notation to name each chord and diameter.

Chords

chords: , ,

Diameters

diameters: , ,

3) Practice: Organizing Information

Fill in the blanks to complete the list.

Diameter Facts

The diameter is twice the length of the of the same circle.

All diameters of a circle have the same .


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A circle has an number of diameters.

The diameter defines the of a circle.

The diameter is the across the center of a circle.


Fill in the blanks to complete the rule and its converse.

5) Practice: Using Visual Cues and Drawing Inferences

Use the diagram to complete the facts and conclusion.


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7) Practice: Using Visual Cues and Drawing Inferences

Use the diagram to complete the facts and conclusions.


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9.2.1 Study: Chords Study Guide

ANSWER KEY


Define each term in 15 words or less.

chord: _________________________________________

_______________________________________________

_______________________________________________

_______________________________________________ (Pages 2 – 3)

Possible response: a line segment that has both endpoints on a circle

diameter: ______________________________________

_______________________________________________

_______________________________________________

_______________________________________________ (Page 22)

Possible response: a chord that passes through the center of a circle


Draw three chords and three diameters on the circles below. Label their endpoints.

Use notation to name each chord and diameter.

Segments will vary. For chords, accept all segments that have both endpoints on the circle. For

diameters, accept all chords that pass through the center of the circle.

Chords (Pages 2 – 3)

chords: , ,

Names will vary. Students should use the two

endpoints of each chord to name it correctly

with notation.

AB; CD; EF

Diameters (Pages 23 – 24)

diameters: , ,

Names will vary. Students should use the two

endpoints of each diameter to name it

correctly with notation.

AB; CD; EF


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3) Practice: Organizing Information (Pages 23 – 24)

Fill in the blanks to complete the list.

Diameter Facts

The diameter is twice the length of the of the same circle.

radius

All diameters of a circle have the same .

length

A circle has an number of diameters.

infinite/endless

The diameter defines the of a circle.

size

The diameter is the across the center of a circle.

distance

4) Practice: Summarizing (Pages 6 – 9)


5) Practice: Using Visual Cues and Drawing Inferences (Pages 6 – 9)

Use the diagram to complete the facts and conclusion.


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Quiz: Congruent Chords Question 1

1a ) Fill in the blank. Given O below, you can conclude that is congruent to ___________.

A.

B.

C. O

D.

Question 2


A.

B.

C.

D. O


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Question 3


A. EF

B. O

C.

D.

Question 4


A. EF

B.

C.

D. O


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Question 5

5a ) What is the length of chord in O below?

A. 5.70 units

B. 5 units

C. 2.5 units

D. 10 units

Question 6

6a ) What is the length of chord in O below?

# Choice

A. 16.04 units

B. 8.02 units

C. 4.01 units

D. 8.31 units


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Question 7

7a ) What is the length of the blue segment in O below?

# Choice

A. 12.48 units

B. 8.53 units

C. 6.24 units

D. 17.06 units

Question 8

8a ) A chord of a circle is a line segment that connects a point on the circle to its center.

A. True

B. False

Question 9

9a ) Which is a true statement about any two chords that are the same distance from the center of a

circle?

A. They are congruent.

B. They are similar.

C. They are parallel.

D. They are perpendicular.


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Question 10

10a ) Which is a true statement about any two congruent chords in a circle?

A. They form an angle.

B. They are perpendicular.

C. They are equidistant from the center of the circle.

D. They are parallel.


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Arcs

This section will introduce you to the part of a circle called an arc. If you visit an arcade for amusement,

it may be in an actual arcade, with an arched semicircular roof overhead.

In Paris, there is a magnificent Arc de Triomphe. In New York's Greenwich Village, there is a wonderful

arch at Washington Square.

The Gateway Arch, designed by Eero Saarien, can be seen from miles away in Saint Louis. The Fort Point

Arch of the Golden Gate Bridge in San Francisco spans 320 feet and consists of four parallel arches.

Architects use arcs (or arches) in the designs of many buildings and sites. In this section, we can learn

about what they are, and build some understanding of their features.

Objectives

Define a circle based on the distance around a circular arc.

Define a central angle, including its relationship to the arc it intercepts.

Define and identify an arc, minor arc, major arc, and semicircle.

Use the proper notation to name an arc.

Apply the properties of arcs and central angles to determine the measures of major and minor

arcs, central angles, circles, and semicircles.


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Arcs

Until now, you mostly worked with straight line segments and points in your studies of circles. But one

of the most striking characteristics of a circle is its curves. This lesson explores the curved parts of

circles, called arcs.

What Is an Arc?

Definition of an Arc

An arc is part of the circumference of a circle. Like a line segment, an arc has a point at each end, called

an endpoint. The distance around a circular arc is like the distance along a line, but it is curved rather

than straight.

Every point along an arc is equidistant from the center of the circle. Making the arc so big that one

endpoint touches the other endpoint means that the arc contains all the possible points that are

equidistant from the center. That's the definition of a circle! The distance around that giant arc is equal

to the circumference of the circle.

What Is a Central Angle?

The picture below shows an arc that is smaller than half the circumference of the circle. Notice how the

two radii form an angle whose sides meet the endpoints of the arc.


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Definition of Central Angle

An angle that has its vertex at the center of a circle is called a central angle.

A central angle is formed by two line segments that connect the center of the circle to the endpoints of

an arc. The central angle intercepts the arc.

Confirm — The Basics of Arcs

You have seen that arcs are parts of the circumference of a circle. You also used these arcs to

form central angles. But do you remember the definition of a central angle?

Major and Minor Arcs

You may have noticed earlier that a central angle actually intercepts two arcs on the circle. One arc is

shorter, and the other is longer.

A minor arc lies between the two sides of a central angle that measures less than 180 . This is the arc

you were experimenting with earlier.

A major arc lies outside of the central angle that is created by a minor arc. The central angle for a major

arc measures more than 180 .

Semicircles

But what if the measure of an arc is exactly 180 ?

When an arc measures exactly half of the circle's circumference, then it is called a semicircle.


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Labeling Minor Arcs

Like with line segments, rays, and lines, you can use notation to represent arcs.

To label arcs, use the symbol shown here: .

For example, the minor arc that joins points A and B below is labeled .

To read this notation out loud, you would say "arc AB."

Labeling Major Arcs

To label the major arc joining A and B, you should use an intermediate point in the label so that you

don't confuse the major and minor arcs.

For example, you can use notation to label the major arc in the picture below. That way, you

would know that the arc goes through point X as it goes from A to B.

Finding the Measure of a Minor Arc

Like angles, arcs can also be measured in terms of degrees. The measure of a minor arc is the measure

of its central angle.

In the example below, the measure of the central angle that intercepts is 90 , so the measure

of is 90 as well.

You can also use this notation to write the measure of the minor arc: .


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Examples:

Finding the Measure of a Major Arc

Finding the measure of the major arc involves one extra step. The measure of a major arc is 360 minus

the measure of the minor arc with the same endpoints.

So, if the endpoints of are separated by a 90 minor arc, then m = 360 – 90 = 270 .


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Finding the Measure of Both Major and Minor Arcs

Can you find the measure of the minor arc and the major arc in the circle below on your own?

Measure of a Circle

Mathematical practice

Using what you know about major arcs and minor arcs, you can show that there are 360 in an

entire circle.

Confirm — Arcs and Their Measures

Over the last set of pages, you saw that an arc is part of the

circumference of a circle. You also used a central angle to

define major arcs and minor arcs. Do you remember

these definitions?

Make sure you also remember the relationship between

the two arc measures. If you need a quick reminder, launch

the tool below.


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9.3.1 Study: Arcs Study Guide

Name:

Date:

Use the questions below to keep track of key concepts from this lesson's study activity.


In 15 words or fewer, explain what an arc is.

arc:


Fill in the blanks to complete the Venn diagram.

Line segment Both Arc

Is part of

a line.

Is

straight.

Have endpoints.

Include all

the between the

endpoints.

Have that can be

measured.

Is part of

a .

Is .


Fill in the blanks to complete the definitions.

A central angle is an angle whose is the center of a circle and

whose are two radii.

A minor arc is an arc created by a central angle.

It lies the two sides of the central angle that intercepts it.

A major arc is an arc created by a central angle.

It lies the two sides of the central angle.

A semicircle is an arc that covers of a circle.


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4) Practice: Monitoring

When reading an arc name, how can you tell the difference between a minor arc and a major arc?


Identify the blue part of each circle. Then use notation to name it.

Identify:

Notation:

Identify:

Notation:

Identify:

Notation:

Identify:

Notation:


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Fill in the blanks to complete the chart.

ARC MEASURES

Arc Measure

minor arc less than

semicircle exactly

major arc more than

whole circle exactly


Fill in the blanks to complete the word equations.

measure of minor arc = measure of angle that intercepts it

measure of minor arc + measure of major arc = measure of

8) Practice: Using Visual Cues

Use the diagram to find each measure.

Name Measure

central angle

minor arc

major arc

circle


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9.3.1 Study: Arcs Study Guide

ANSWER KEY

1) Practice: Summarizing (Page 3)

In 15 words or fewer, explain what an arc is.

arc:

Possible response: An arc is part of the circumference of a circle.

2) Practice: Organizing Information (Page 3)

Fill in the blanks to complete the Venn diagram.

Line segment Both Arc

Is part of

a line.

Is

straight.

Have endpoints.

two

Include all

the between the

endpoints.

points

Have that can be

measured.

length

Is part of

a .

circle

Is .

curved


Fill in the blanks to complete the definitions.

A central angle is an angle whose is the center of a circle and

whose are two radii.

vertex; sides (Page 5)

A minor arc is an arc created by a central angle.

It lies the two sides of the central angle that intercepts it.

between/inside (Page 8)


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A major arc is an arc created by a central angle.

It lies the two sides of the central angle.

outside (Page 8)

A semicircle is an arc that covers of a circle.

half (Page 9)

4) Practice: Monitoring (Pages 10 – 11)

When reading an arc name, how can you tell the difference between a minor arc and a major arc?

Possible response: A minor arc name has 2 points (its 2 endpoints). A major arc name has 3 points (its 2

endpoints and 1 point between them).

5) Practice: Using Visual Cues and Summarizing (Pages 12 – 15)

Identify the blue part of each circle. Then use notation to name it.

Identify:

central angle

Notation:

∠ACB

Identify:

minor arc

Notation:


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Identify:

major arc

Notation:

Identify:

semicircle

Notation:


Fill in the blanks to complete the chart.

ARC MEASURES

Arc Measure

minor arc (Page 8) less than

180

semicircle (Page 9) exactly

180

major arc (Page 8) more than

180

whole circle (Page 15) exactly

360


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Fill in the blanks to complete the word equations.

measure of minor arc = measure of angle that intercepts it

central

measure of minor arc + measure of major arc = measure of

circle

8) Practice: Using Visual Cues (Pages 12 – 15)

Use the diagram to find each measure.

Name Measure

central angle

120°

minor arc

120°

major arc

240°

circle

360°


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Quiz: Arc Types and Measure Question 1

1a ) Given O below, is a minor arc, a major arc, or a semicircle?

A. Minor arc

B. Semicircle

C. Major arc

Question 2


A. Semicircle

B. Minor arc

C. Major arc


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Question 3


A. Semicircle

B. Minor arc

C. Major arc

Question 4

4a ) What is the measure of in O below?

A. 260

B. 100

C. 80

D. 280


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Question 5

5a ) What is the measure of in O below?

A. 240

B. 60

C. 120

D. 300

Question 6

6a ) An angle whose vertex is at the center of a circle is a middle angle of that circle.

A. True

B. False

Question 7

7a ) The sides of a central angle are two ______ of the circle.

A. radii

B. chords

C. arcs

D. diameters


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Question 8

8a )An arc that lies between the two sides of a central angle is called a ________.

A. major arc

B. minor arc

C. semicircle

Question 9

9a ) An arc that measures 180 is called a ________.

A. major arc

B. semicircle

C. minor arc

Question 10

10a ) An arc that lies outside of a central angle is called a _______.

A. minor arc

B. major arc

C. semicircle

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