Copyright © 2014 Pearson Education, Inc. 6 CHAPTER 6.3 Tangents Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Congruent Arcs Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Congruent Arcs It is possible for two arcs of different circles to have the same measure but different lengths. It is also possible for two arcs of different circles to have the same length but different measures. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Congruent Arcs Arcs not congruent: • same measure • not same or congruent circles, so different lengths • same lengths • not same or congruent circles, so different measures Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Tangents A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. The point where a circle and a tangent intersect is the point of tangency. is a tangent line. is a tangent ray. is a tangent segment. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Postulate 6.3 If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Postulate 6.4 If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency. Copyright © 2014 Pearson Education, Inc.

Finding Angle Measures Multiple Choice and are tangent to circle O. What is the value of x? a. 58 b. 63 c. 90 d. 117 Solution A tangent line is perpendicular to the radius at the point of tangency. The two given segments are tangent to the circle. Angles L and N are right angles. LMNO is a quadrilateral. The sum of the angles is 360 degrees. Copyright © 2014 Pearson Education, Inc.

Finding Angle Measures m∠L + m∠M + m∠N + m∠O = 360° 90° + m∠M + 90° + 117° = 360° 297° + m∠M = 360° m∠M = 63° Since m∠M = 63°, or x°, then x = 63. The correct answer is b. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Finding a Radius What is the radius of circle C? Solution Use the Pythagorean Theorem to find x. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 6.19 If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Construction Construction 6.1 Construction 6.2 Page 299 Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 6.16 The angle formed by a tangent and a chord has a measure one-half the measure of it’s intercepted arc. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 6.17 The angle formed by the intersection of a tangent and a secant has measure one-half the difference of the measures of the intercepted arcs. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 6.18 The angle formed by the intersection of two tangents has a measure one half the distance of the intercepted arcs. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 6.20 If a secant and a tangent are drawn to a circle from an external point, the length of the tangent segment is the geometric mean between the length of the secant and its external segment. Copyright © 2014 Pearson Education, Inc.

Finding Tangent Segment Lengths If are tangents to circle D, find the value of x. Solution From Theorem 12.1-3, we know that AC ≅ AB. The value of x is 9 or 1. The check is left to you. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Tangents Two circles in a plane can share a tangent line. Common tangents are lines or segments or rays that are tangent to more than one circle. Copyright © 2014 Pearson Education, Inc.

Possible Intersections of Two Circles Two Points of Intersection 2 Common Tangents 2 external tangents (green) 0 internal tangents Copyright © 2014 Pearson Education, Inc.

Possible Intersections of Two Circles One Point of Intersection 3 Common Tangents 2 externally tangent circles 2 external tangents (green) 1 internal tangent (black) Copyright © 2014 Pearson Education, Inc.

Possible Intersections of Two Circles One Point of Intersection 1 Common Tangent 2 internally tangent circles 1 external tangent (green) 0 internal tangent Copyright © 2014 Pearson Education, Inc.

Possible Intersections of Two Circles No Points of Intersection 0 Common Tangents 2 concentric circles one circle floating inside the other, without touching 0 external tangents 0 external tangents 0 internal tangents 0 internal tangents Copyright © 2014 Pearson Education, Inc.

Possible Intersections of Two Circles No Points of Intersection 4 Common Tangents 2 external tangents (green) 2 internal tangents (black) Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Tangents The line that passes through the centers of two circles is called their line of centers. If two circles are tangent, internally or externally (see middle examples above), then their common point of tangency is on their line of centers, and these circles are called tangent circles. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Constructions Page 307 Construction 6.3 Construction 6.4 Copyright © 2014 Pearson Education, Inc.