Warm-Up1. Calculate the exact value of sine and

cosine of 30°

2. Calculate the sum of the square of the sine and cosine of 30°

3. Explain what you think you would get if you did the same thing (find the sine and cosine of the angle, square them, and add them together) with 60°, 45°, or any other angle

Circles and Spheres Key Standards

MM2G3. Students will understand the properties of circles.

b. Understand and use properties of central, inscribed, and related angles.

 

CircleWhat is the definition of a circle?

A circle is the locus of points that are a constant distance from a given point, called the center.

The circle is named for its center, ex PWhat 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

Locus of PointsLook at the investigation on page 460 –

461 of the Geometry book.

Central AngleTwo radii form a central angleA central angle of a circle is an angle

whose vertex is the center of the circle.

ChordsA chord is a segment whose endpoints

are on a circleA diameter is a chord what contains the

center of the circle.

ArcsAn 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

ArcsThe measure of a major arc is defined as

the difference between 360 and the measure of its associated minor arc.

Major arcs and semicircles are named by their end points and a point on the arc

Major arc: Central angle > 180

Semicircle: Central angle = 180

NomenclaturePay 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

•Example 1:

60 60

Central Angle = APB

Minor arc = AB mAB = mAPB = 60

Major arc = ACB mACB = mACB = 360 - 60 = 300

Minor arcMajorarc

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 PostulateAdjacent 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 = mAB + mBC

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

BAB and are in the

same circle and m = m = 45°. So, =

DC

ABDCDC

AB

45°

45°

Q

S

P

R

Ex. 4: Identifying Congruent Arcs

Find the measures of the blue arcs. Are the arcs congruent?

RSPQ and are in

congruent circles and m = m = 80°.

So, =

PQ

RSRS

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

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°

PracticePage 193, # 3 – 39 by 3’s and 19

(14 problems)