CIRCLES

Chapter 10

Tangents to Circles lesson 10.1

California State Standards7: Prove and Use theorems involving properties of circles.21: Prove and Solve relationships among chords, secants and tangents.

definitions

CircleThe set of all points in a plane that are equidistant

from a given point.

CenterThe given point.

RadiusA line segment with the center as one endpoint and

a point on the circle as the other endpoint.The distance from the center to a point on the circle.

CircleThe set of all points in a plane that are equidistant

from a given point.

CenterThe given point.

RadiusA line segment with the center as one endpoint and

a point on the circle as the other endpoint.The distance from the center to a point on the circle.

definitions

C

P

the plural of radius is radii

a circle is namedby its center

C

is a radius is the length

CPCP

definitions

Congruent CirclesCircles with the same radius

DiameterA line segment with endpoints on the circle

that contains the center of the circle.The distance across a circle through the center.

2r = d

definitions is a diameterRPCongruent Circles

Circles with the same radius

DiameterA line segment with endpoints on the circle that contains the center of the circle.The distance across a circle through the center. 2r = d

PCR

C R CR RC

2r = d

definitions

ChordA segment whose endpoints are on the circle.A diameter is a “specialized” chord.

SecantA line that intersects a circle in two points.

TangentA line that intersects a circle in exactly one point.The circle and line must lie in the same plane.

definitions

ChordA segment whose endpoints are on the circle.A diameter is a “specialized” chord.

SecantA line that intersects a circle in two points.

TangentA line that intersects a circle in exactly one point.The circle and line must lie in the same plane.

CA

B is a chordAB

F

GFG is a secant,------------- -

T

TU is a tangent,------------- - U

radiusC

P

A

B

Q

X

radiu

s

radiu

sdia

met

er chord

secant

tangent

Identify each line or segment

chord

definition

Tangent CirclesCoplanar circles that intersect in exactly one point.

externally tangent circles

definition

Tangent CirclesCoplanar circles that intersect in exactly one point.

internally tangent circles

definition

Concentric CirclesCoplanar circles with a common center.

definition

Common TangentA line or segment that is tangent to two

coplanar circles• Common Internal Tangent crosses between the circles• Common External Tangent stays along the edges of the

circles

Common TangentA line or segment that is tangent to two

coplanar circles• Common Internal Tangent crosses between the circles• Common External Tangent stays along the edges of the

circles

definitionCommon Internal Tangent

Common External Tangent

example

Is the common tangentinternal or external?

external

C DT

example

C

H

B

A I

D

E

F G

Describe each segment

tangent

diameter

chord

radius

secant

AH

EI

DF

CE

HG

definitions

exterior

interior

Interior of a CircleThe set of points inside the circle

Exterior of a CircleThe set of points outside the circle.

theorem

Circle Tangent-Radius PerpendicularIf a line is tangent to a circle,then it is perpendicular to the radius drawn to the point of tangency.

t

C

P

t CP

C

theorem

Circle Tangent-Radius Perpendicular ConverseIf a line is perpendicular to a radius of a circle at the endpoint on the circle,then the line is tangent to the circle.

tP

is tangent to t C

example

C

A

B

43

45

11

Is tangent to ?AB C2 2 245 ? 43 11

2025 ?1846 1212025 1970

is not to No

AB CB

10r 32 320r

32 256 576r

2 232 256 576r r r

2 2 2( 16) 24r r

example

C

AB

r

r 16

24

is tangent to .

Find the radius of the circle.

AB C

theorem

Congruent TangentsIf two segments from the same exteriorpoint are tangent to a circle,then the segments are congruent.

P

C

Q

S

is tangent to SP C

is tangent to SQ C

SQ SP

2 25x

example

A

B

D

x2 – 4

21

C5x

2 4 21x

and are tangent to .

Find the value of x.

AB AD C

example

B

D

x

36

15

yz

A

C

Find the value of , , and .x y z

AB AD

36x

36

15

CB CD15y

Find the value of , , and .x y z

36

example

B

D

36

15

z

A

C 39z

2 1521z

2 225 1296z

2 2 215 36z

39

15

exampleWhat are the coordinates of each center?

What is the radius of each circle?

x

y

A B

(2,2)A (6,2)B

2A r 2B r

A B

exampleDescribe the intersection of the two circles.

x

y

A B

and are externally tangent

A B

exampleDescribe the common tangents of the circles.

x

y

A B

and have anintenal tangent at 4

A Bx

They have 2 external tangents.

4y 0y

A

y

example

B

What are the coordinates of each center? (0,1)A (2,1)B

What is the radius of each circle? 1A r 3B r

Describe any common tangents.

x _1 is a common external tangent

x

5. SSS 4. reflexive

3. def. radius

2. Tangents Th

5. ABC ADC 4. CA CA

3. CB CD

2. AB AD

1. and are tangent to

AB ADC

Statement

Given: and are tangent to Prove:

AB AD CABC ADC

C

AB

D

Reason

1. given