Chapter 9Circles

• Define a circle and a sphere.

• Apply the theorems that relate tangents, chords and radii.

• Define and apply the properties of central angles and arcs.

Bring a Compass Tomorrow

9.1 Basic Terms

Objectives

• Define and apply the terms that describe a circle.

The Circle

is a set of points in a plane equidistant from a given point.

A

B

The Circle

The given distance is a radius (plural radii)

A

B

radius

The Circle

The given point is the center

A

B

radius

center

The Circle

A

BPoint on circle

Chord

any segment whose endpoints are on the circle.

A

BC

chord

Diameter

A chord that contains the center of the circle

A

BC

diameter

any line that contains a chord of a circle.

Secant

A

BC

secant

Tangent

any line that contains exactly one point on the circle.

A

B

tangent

Point of Tangency

A

BPoint of tangency

Sphere

is the set of all points equidistant from a given point.

AB

Sphere

Radii

Diameter

Chord

Secant

TangentA

B

D

C

E

F

Congruent Circles (or Spheres)

have equal radii.

A D

BE

Concentric Circles (or Spheres)

share the same center.

O

G

Q

Inscribed/Circumscribed

A polygon is inscribed in a circle and the circle is circumscribed about the polygon if each vertex of the polygon lies on the circle.

P

M

Q

O

N

R

L

Name each segment

P

M

Q

O

N

R

L

OM

P

M

Q

O

N

R

L

MN

P

M

Q

O

N

R

L

MN

P

M

Q

O

N

R

L

MQ

P

M

Q

O

N

R

L

ML

P

M

Q

O

N

R

L

ML

P

M

Q

O

N

R

L

Point M

9.2 Tangents

Objectives

• Apply the theorems that relate tangents and radii

TheoremIf a line is tangent to a circle, then the line is perpendicular to the radius

drawn to the point of tangency.

A

B

tangent

C

90m ABC Sketch

Corollary

Tangents to a circle from a common point are congruent.

A

X

Y

ZXY XZSketch

tangent

tangent

Theorem

If a line in the plane of a circle is perpendicular to a radius at its endpoint, then the line is a tangent to the circle.

AX

B

tangent

Inscribed/Circumscribed

When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon.

White Board Practice

Common Tangents

are lines tangent to more than one coplanar circle.

A

X

B

tangentR

Common External Tangents

A

XB

R

Common External Tangents

A

X

B

R

Common Internal Tangents

A

X

B

R

Common Internal Tangents

A

X

B

R

Construction 8Given a point on a circle, construct the tangent to the circle through the point.

Given:

Construct:

Steps:

with point A Btangent line l to through A B

Remote Time

• How many common external tangents can be drawn?

Remote Time

• How many common external tangents can be drawn?

Remote Time

• How many common external tangents can be drawn?

Remote Time

• How many common external tangents can be drawn?

Remote Time

• How many common external tangents can be drawn?

Remote Time

• How many common external tangents can be drawn?

Remote Time

• How many common internal tangents can be drawn?

Remote Time

• How many common internal tangents can be drawn?

Remote Time

• How many common internal tangents can be drawn?

Remote Time

• How many common internal tangents can be drawn?

Remote Time

• How many common internal tangents can be drawn?

Remote Time

• How many common internal tangents can be drawn?

Tangent Circles

are circles that are tangent to each other.

A

B

R

Externally Tangent Circles

A

B

R

Internally Tangent Circles

A

B

R

Remote Time

• Are the circlesA. Externally Tangent

B. Internally Tangent

C. None

Remote Time

• Are the circlesA.Externally Tangent

B.Internally Tangent

C.None

Remote Time

• Are the circlesA.Externally Tangent

B.Internally Tangent

C.None

Remote Time

• Are the circlesA.Externally Tangent

B.Internally Tangent

C.None

Remote Time

• Are the circlesA.Externally Tangent

B.Internally Tangent

C.None

Remote Time

• Are the circlesA.Externally Tangent

B.Internally Tangent

C.None

9.3 Arcs and Central Angles

Objectives

• Define and apply the properties of arcs and central angles.

Central Angle

is formed by two radii, with the center of the circle as the vertex.

B

A C

Arc

an arc is part of a circle like a segment is part of a line.

B

AC

AC

Arc Measure

the measure of an arc is given by the measure of its central angle.

B

AC

80

80

AC

80mAC

Minor Arc

an unbroken part of a circle with a measure less than 180°.

B

AC

AC

Semicircle

an unbroken part of a circle that shares endpoints with a diameter.

B

A C

Major Arc

an unbroken part of a circle with a measure greater than 180°.

BA C

D

ACD

Adjacent Arcs

arcs that have exactly one point in common.

B

A C

D

AD DC

Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the sum of the arcs.

B

A C

D

Sketch

mADCmDCmAD

Congruent Arcs

arcs in the same circle or in congruent circles that have the same measure.

B

A C

D90

90

DCAD

mDCmAD

White Board Practice

Name two minor arcs

R

C

SA

O

White Board Practice

AR, RC, RS, AS, SC

R

C

SA

O

White Board Practice

Name two major arcs

R

C

SA

O

White Board Practice

ARS, ACR, RCS, RSA, RSC, CRS, CSR

R

C

SA

O

White Board Practice

Name two semicircles

R

C

SA

O

White Board Practice

ARC, ASC

R

C

SA

O

White Board Practice

Name an acute central angle

R

C

SA

O

White Board Practice

AOR

R

C

SA

O

Theorem

In the same circle or in congruent circles, two minor arcs are congruent only if their central angles are congruent.

B

A C

D

90 90DCAD

DBCABD

White Board Practice

Name two congruent arcs

R

C

SA

O

White Board Practice

ARC, ASC

R

C

SA

O

Group Practice

• Give the measure of each arc.

4x

3x 3x + 10

2x

2x-1

4

A

B

C

D

E

Group Practice

m AB = 88

m BC = 52

m CD = 38

m DE = 104

m EA = 784x

3x 3x + 10

2x

2x-1

4

A

B

C

D

E

The radius of the Earth is about 6400 km.

6400

6400

O

BA

The latitude of the Arctic Circle is 66.6º North. That means the m BE 66.6º.

6400

6400

O

BA

EW

66.6º

Find the radius of the Arctic Circle

6400

O

BA

EW

66.6º

xº

Find the radius of the Arctic Circle

6400

O

BA

EW

66.6º

23.4º

Lecture 4 (9-4)

Objectives

• Define the relationships between arcs and chords.

Chord of the ArcThe minor arc between the endpoints of a

chord is called the arc of the chord, and the chord between the endpoints of an arc is the chord of the arc.

BA

D

Theorem 9-4

Sketch

In the same circle or in congruent circles, congruent arc have congruent chords and congruent chords have congruent arcs.

B AC

D

BD DC

BD DC

Theorem 9-5

Sketch

A diameter that is perpendicular to a chord bisects the chord and its arc.

B

AC

DX

Y

DC BY

DX XC

DY YC

Theorem 9-6

Sketch

In the same circle or in congruent circles, chords are equally distant from the center only if they are congruent.

B

AC

D

X

YA XA

BD EC

Y

E

9.5 Inscribed Angles

Objectives• Solve problems and

prove statements about inscribed angles.

• Solve problems and prove statements about angles formed by chords, secants and tangents.

Inscribed Angle

B

A

C

An angle formed by two chords or secant lines whose vertex lies on the circle.

Theorem

B

A

C

The measure of an inscribed angle is half the measure of the intercepted arc.

mACABCm2

1

Corollary

B

A

C

If two inscribed angles intercept the same arc, then they are congruent.

ABC ADC

Sketch

D

Corollary

C

A

An angle inscribed in a semicircle is a right angle.

90m ABC

B

O

Corollary

C

A

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

B

O D180

180

m A m C

m B m D

An angle formed by a chord and a tangent has a measure equal to half of the intercepted arc.

Theorem

C

A

B

O

D

mADBABCm2

1

Construction 9Given a point outside a circle, construct the tangent to the circle through the point.

Given:

Construct:

Steps:

with point A Btangent line l to through A B

9.6 Other Angles

Objectives

• Solve problems and prove statements involving angles formed by chords, secants and tangents.

TheoremThe angle formed by two intersecting chords

is equal to half the sum of the intercepted arcs.

A

D

B

C

E

1)(

2

11 mDEmCBm

TheoremThe angle formed by secants or tangents with the

vertex outside the circle has a measure equal to half the difference of the intercepted arcs.

A

D

B

CE

1

F

)(2

11 mEFmBDm

AO

G

F

D

E

CB

123

45

6

7

8

AB is tangent to circle O.AF is a diameterm AG = 100m CE = 30m EF = 25

9.7 Circles and Lengths of Segments

Objectives

• Solve problems about the lengths of chords, secants and tangents.

TheoremWhen two chords intersect, the product of

their segments is equal.

A

D

B

XE

F

XBFXXDEX

TheoremWhen two secant segments are drawn to a circle

from a common point, the product of their length times their external segments is equal.

A

D

B

CE

1

F

CFCDCECB

Whole Piece Outside Piece = Whole Piece Outside Piece

TheoremWhen a secant and a tangent are drawn from a

common point, the product of the secant and its external segment is equal to the tangent squared.

A

D

C

E

FCECECFCD

Whole Piece Outside Piece = Whole Piece Outside Piece