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Circles and Spheres Key

Standards

MM2G3. Students will understand the

properties of circles.

b. Understand and use properties of central,

inscribed, and related angles.

Locus of Points

Look at the investigation on page 460 –

461 of the Geometry book.

Investigate: Given a point A, what is the

locus of points in a plane that are 4

inches from point A?

Method: Locate (fold paper hotdog and

hamburger) and mark the center of a

piece of paper. Use a ruler, a piece of

paper, and a radius of 4 inches.

Circle What is the definition of a circle?

A circle is the locus of points, in a plane,

that are a constant distance from a given

point, called the center.

The circle is named for its center, ex P

What is that constant distance called?

A radius is a segment whose endpoints are

the center and any point on the circle.

How many radii does circle have?

An infinite number

Central Angle

Two radii form a central angle

A central angle of a circle is an angle

whose vertex is the center of the circle.

Chords

A chord is a segment whose endpoints

are on a circle

A diameter is a chord that contains the

center of the circle.

Secants

A secant is a line that intersects the

circle at two points

A secant that included the center also

includes the diameter

Tangent

A tangent is a line that intersects the

circle only once

A tangent is always perpendicular to a

radius at the point of tangency

Arcs An arc is an unbroken part of a circle.

Minor Arcs are named for their end points.

The measure of a minor arc is defined to

be the measure of its central angle.

Minor arc: Central angle < 180

Arcs The measure of a major arc is defined as

the difference between 360 and the

measure of its associated minor arc.

Semicircle: Central angle = 180

Major arc: Central angle > 180

Major arcs and semicircles are named by

their end points and a point on the arc

Nomenclature

Pay particular attention to the

nomenclature as shown in the following

slide.

The arc AB is designated:

This same nomenclature will be used to

designate the length of the arc later.

The measure of the arc in degrees is

designated:

AB

ABm

Warm-Up

Draw, name and label a:

• Circle

• Chord

• Secant

• Tangent

• Radius

• Diameter

• Central Angle

• Minor arc

• Major Arc

• Point of

Tangency

•Example 1:

60 60

Central Angle = APB

Minor arc = AB mAB = mAPB = 60

Major arc = ACB mACB = mACB =

360 - 60 = 300

Minor arcMajor

arc

C

P

B

A

Ex. 2: Finding Measures of Arcs

Find the measure

of each arc of R.

a.

b.

c.

MNMPN

PMN PR

M

N80°

Ex. 2: Finding Measures of Arcs

Find the measure

of each arc of R.

a.

b.

c.

Solution:

is a minor arc, so

m = mMRN

= 80°

MNMPN

PMN PR

M

N80°

MN

MN

Ex. 2: Finding Measures of Arcs

Find the measure

of each arc of R.

a.

b.

c.

Solution:

is a major arc, so

m = 360° – 80°

= 280°

MNMPN

PMN PR

M

N80°

MPN

MPN

Ex. 2: Finding Measures of Arcs

Find the measure

of each arc of R.

a.

b.

c.

Solution:

is a semicircle, so

m = 180°

MNMPN

PMN PR

M

N80°

PMN

PMN

Arc Addition Postulate Adjacent arcs have exactly one point in

common.

The measure of an arc formed by two

adjacent arcs is the sum

of the measures

of the two arcs

m ABC = m AB + m BC

B

C

A

Ex. 3: Finding Measures of Arcs

Find the measure of

each arc.

a.

b.

c.

m = m + m =

40° + 80° = 120°

GE

GEFR

EF

G

H

GFGE

GH

HE

40°

80°

110°

Ex. 3: Finding Measures of Arcs

Find the measure of

each arc.

a.

b.

c.

m = m + m =

120° + 110° = 230°

GE

GEFR

EF

G

H

GF

EF

40°

80°

110° GEF

GE

Ex. 3: Finding Measures of Arcs

Find the measure of

each arc.

a.

b.

c.

m = 360° - m =

360° - 230° = 130°

GE

GEFR

EF

G

H

GF

40°

80°

110° GF

GEF

W X

40

Q

40

Z Y

Congruent Arcs

In a circle or in congruent circles, two

minor arcs are congruent iff their

corresponding central angles are

congruent.

Need Congruent:

Central angles

Radii.

Ex. 4: Identifying Congruent Arcs

Find the measures

of the blue arcs.

Are the arcs

congruent?

C

D

A

B

AB and are in the

same circle and

m = m = 45°.

So, =

DC

AB

DCDC

AB

45°

45°

Q

S

P

R

Ex. 4: Identifying Congruent Arcs

Find the measures

of the blue arcs.

Are the arcs

congruent? RS

PQ and are in

congruent circles and

m = m = 80°.

So, =

PQ

RS

RS

PQ

80°

80°

X

W

Y

Z

Ex. 4: Identifying Congruent Arcs Find the measures of

the blue arcs. Are the arcs congruent?

65°

m = m = 65°, but

and are not arcs of the

same circle or of

congruent circles, so

and are NOT

congruent.

XY

ZW

XY

ZW

XY

ZW

Practice

Page 193, # 3 – 39 by 3’s and 19

(14 problems)

Warm-Up: Identify Diameter

Chord

3 Radii

m AB

m CAB

Name of circle

3 minor arcs

3 major arcs

Semicircle

mBSC

mBSA = 40.00

mCSD = 25.00

B

A

C

S

D

F E

Application:

Determine each central angles to make

a pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 25

Orange 15

Green 10

Application:

Determine each central angles to make

a pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 25

Orange 15

Green 10

Total 50

Application:

Determine each central angles to make

a pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 25 50

Orange 15 30

Green 10 20

Total 50

Application:

Determine each central angles to make

a pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 25 50

Orange 15 30

Green 10 20

Total 50 100

Application:

Determine each central angles to make

a pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 25 50 180

Orange 15 30 108

Green 10 20 72

Total 50 100

Application:

Determine each central angles to make

a pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 25 50 180

Orange 15 30 108

Green 10 20 72

Total 50 100 360

Application:

What is the central angles if we wanted

to combine Blue and Green?

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 25 50 180

Orange 15 30 108

Green 10 20 72

Total 50 100 360

252°

Class Work – In Groups

Determine each central angles to make a

pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 8

Orange 7

Green 5

Class Work – In Groups

Determine each central angles to make a

pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 8

Orange 7

Green 5

20

Class Work – In Groups

Determine each central angles to make a

pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 8 40

Orange 7 35

Green 5 25

Total 20 100

Class Work – In Groups

Determine each central angles to make a

pie chart from the following data:

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 8 40 144

Orange 7 35 126

Green 5 25 90

Total 20 100 360

Class Work – In Groups

What would the central angle be if we

combined the Blue and Green?

Category Number of

each color

% Number of

Degrees in the

Central Angle

Blue 8 40 144

Orange 7 35 126

Green 5 25 90

Total 20 100 360

234°

40%

35%

25%

According to the 2007-2008

Pet Owners survey:[

Animal # Households

with a Pet

(millions)

# Pets

(millions)

Bird 6.0 15.0

Cat 38.2 93.6

Dog 45.6 77.5

Equine 3.9 13.3

Freshwater fish 13.3 171.7

Saltwater fish 0.7 11.2

Reptile 4.7 13.6

Small pets 5.3 15.9

Homework

Problem18 of 11 – 2 Practice

Problems 19 – 27 of Exercises Handout

page 456

Homework:

11 – 3 Study Guide