6.1 (M2) Use Properties of Tangents

Vocabulary

• Circle- the set of all pts in a plane that are equidistant from a given pt. called the center of the circle.

• Radius- 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.

• Two polygons are similar if corresponding angles are congruent and corresponding side lengths are proportional. ΔABC ΔDEF

Chord• Chord- a segment whose endpoints are on

the circle.

A B

P

P is the center of the circle

A

BSegment AB is a diameter

C

Segments AP, PB, and PC are radii

Secant

• Secant- a line that intersects a circle in 2 pts

A

B

Tangent

• Tangent- a line in the plane of the circle that intersects the circle in exactly one point, called the point of tangency.

Exampletell whether the segment is best described as a chord, secant,

tangent, diameter or radius• Segment AH

• Segment EI

• Segment DF

• Segment CE

A

B

C

D

E

F

G

H

I

tangentDiameter

Chord

radius

More Definitions• Tangent circles- circles that intersect in

one pt• Concentric circles- circles that have a

common center but different radii lengths

• Common tangent- a line or segment that is tangent to two circles

• Common internal tangent- a tangent that intersects the segment that connects the centers of the circles

• Common external tangent- does not intersect the segment that connects the centers

Tangent Circles

Concentric Circles

Common Internal Tangent

Common External Tangent

ExampleCommon internal or external

tangent?

external

Theorem• In a plane, a line is tangent to a circle if

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

ExampleIs segment CE tangent to circle D?

Explain

D

E

C

11

45

43

112+432=452

121+1849=20251970=2050

NO

Examplesolve for the radius, r

A

B

Cr

r28ft

14ft

r2+282=(r+14)2

r2+ 784=r2+ 28r+196784=28r+196

588=28r21=r

Theorem

• Tangent segments from a common external point are congruent.

Examplesegment AB is tangent to circle C at pt B. segment AD is tangent to

circle C at pt D. Find the value of X

C

B

D

A

x2+8

44

x2+8=44x2=36

X=6