Section 9-2

Tangents

Theorem 9-1: If a line is tangent to a circle, then the line is

perpendicular to the radius drawn to the point of tangency.

If AB is tangent to Circle Q at point C, then QC

AB.A BC

Q

Q is the center of the circle. C is a point of tangency.

Example: Given Circle Q with a radius length of 7. D is a point of tangency. DF = 24, find the length of QF.

FD

Q

7

24

72 + 242 = QF2

QF = 25

Extension: Find GF.

G

QG = 7QF = 25 GF = 18

NOTE: G is NOT

necessarily the midpoint of

QF!!

Theorem 9-2: If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the

circle.

This is the converse of Theorem 9-1.

Common Tangent – a line that is tangent to two coplanar

circles.Common Internal Tangent

Intersects the segment joining the centers.

Common External Tangent

Does not intersect the segment joining the

centers.

Tangent Circles – coplanar circles that are tangent to the same line at the same point.

Internally Tangent Circles Externally

Tangent Circles

Section 9-3: Arcs & Central Angles

Definition: a Central Angle is an angle with its vertex at the

center of the circle.

A

O B

AOB is a central angle of circle O.

A

O B

This central angle intercepts an arc of circle O. The intercepted

arc is AB.

Arcs are measured in degrees, like

angles. The measure of the

intercepted arc of a central angle is

equal to the measure of the central angle.

110°

110°

Types of arcs:

O

AD

C

**Major Arcs and Semicircles are ALWAYS named with 3 letters.**

Example: AD

Example: ACD

Example: ADC

Major Arc – measures more than 180°

Semicircle – measures exactly 180°

Minor Arc – measures less than 180°

Adjacent Arcs – Two arcs that share a common endpoint, but

do not overlap.

AE

O

F AF and FE are adjacent arcs.

EF and FAE are adjacent arcs.

Name…

1.Two minor arcs

2.Two major arcs

3.Two semicircles

4.Two adjacent arcs

V

W Z

Y

Circle Z

X

VW, WY

VYW, XYV , WVY

VWY, VXY

VW & WY or YXV & VW

Give the measure of each angle or arc.

OZ

T

W

XY

50°

30°

1. WOT =

100°

50°

2. mWX =100°

3. mYZ = 90°

4. mYZX= 330°

5. mXYT =210°

6. mWYZ= 220°

7. mWZ =140°

50°