Chapter 10Properties of Circles

10.1 Using Properties of TangentsCircle- a set of all points in a

plane that are equidistant from a given point called the center

C

Radius- a segment whose endpoints are the center and any point on the circle

Chord- a segment whose endpoints are on a circle

Diameter- a chord that contains the center of the circle

Secant- a line that intersects a circle in two points

Tangent- a line in the plane of a circle that intersects the circle in exactly one point

Can you name it?ChordRadiusDiameterSecantTangentPoint of Tangency

C

E

D

B

J

A

G

F

H

I

Coplanar circlesConcentric circles

Internally tangent circles

Externally tangent circles

Common tangentsInternal common tangent

External common tangent

TheoremsIn a plane, a line is tangent to a

circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle

mQ

P

Tangent segments from a common external point are congruent.

STRSP

T

S

R

ExamplesIs segment BC tangent to circle A

if segment AB is a radius?

6725

60

A

B C

ExampleS is a point of tangency. Find r.

r36r

48

T

S R

ExamplePoint R and T are tangent to

circle P. Find x.

6x-8

25

P

T

S

R

ExampleHow many common tangents?

B.

A.C.

10.2 Finding Arc MeasuresCentral angle- an angle whose

vertex is the center of the circle

Major arc

Minor arc

Semicircle

80C

A

D

BE

Arc Addition PostulateThe measure of an arc formed by

two adjacent arcs is the sum of the measures of the two arcs.

C

A

D

B

E

Congruent Circles and ArcsTwo circles are congruent if they

have the same radius.Two arcs are congruent if they

have the same measure and they are arcs of the same circle or of congruent circles.

The radii of a circle, or of congruent circles, are congruent.

Examples

Find the following:

AE

ABD

EBD

135C

A

D

B

E

ExampleAre arcs AB and DE congruent?

A.

45

45C

A

D

B

E

B.

E

C

A

D

B

C.

C

A

B

D

E

ExampleAges of people in a town (in

years)

mGA

mGE

mGFE

mBFA60 80

90

30

100C

A

E

G

F

B

10.3 Applying Properties of ChordsIn the same circle, or in

congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Example: Find the measure of arc SR.

100

P Q

R

S T

U

If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

If the diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Example

FindBFCFAF

10

12

F

C

A

E

D B

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

A

B C

D

E

ExampleBC= 2x +6ED = 3x – 1 Find BC

A

B C

D

E

ExampleThree props are placed on a

stage (P,Q,R). Where do you put the table so that it is the same distance from each prop?

P

Q

R