Section 9-2 Tangents

If AB is tangent to Circle Q at point C, 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. Q is the center of the circle. C is a point of tangency. If AB is tangent to Circle Q at point C, Q then QC ^ AB. A C B

NOTE: G is NOT necessarily the midpoint of QF!! Example: Given Circle Q with a radius length of 7. D is a point of tangency. DF = 24, find the length of QF. 72 + 242 = QF2 F D Q 7 24 QF = 25 G NOTE: G is NOT necessarily the midpoint of QF!! Extension: Find GF. QF = 25 QG = 7 GF = 18

This is the converse of Theorem 9-1. 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

ÐAOB is a central angle of circle O. Definition: a Central Angle is an angle with its vertex at the center of the circle.

This central angle intercepts an arc of circle O. The intercepted 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° B O This central angle intercepts an arc of circle O. The intercepted arc is AB.

**Major Arcs and Semicircles are ALWAYS named with 3 letters.** Types of arcs: C Minor Arc – measures less than 180° O A Example: AD D Major Arc – measures more than 180° Example: ACD Semicircle – measures exactly 180° Example: ADC **Major Arcs and Semicircles are ALWAYS named with 3 letters.**

Adjacent Arcs – Two arcs that share a common endpoint, but do not overlap. AF and FE are adjacent arcs. F E EF and FAE are adjacent arcs. A O

Y W Z X V Name… Two minor arcs 2.Two major arcs VW, WY 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

Y X Z O W T 50° Give the measure of each angle or arc. 1. mÐWOT = 50° 2. mWX = 100° 30° 3. mYZ = 90° Z 100° 4. mYZX = 330° O 50° 5. mXYT = 210° W 6. mWYZ = 220° 50° T 7. mWZ = 140°