CIRCLES Chapter 10

Tangents to Circles lesson 10.1 Objective: 1. Identify segments and lines related to a circle. 2. Use properties of a tangent to a circle.

definitions Circle Center Radius The set of all points in a plane that are equidistant from a given point. Center The given point. Radius A line segment with the center as one endpoint and a point on the circle as the other endpoint. The distance from the center to a point on the circle.

the plural of radius is radii definitions P Circle The set of all points in a plane that are equidistant from a given point. Center The given point. Radius A line segment with the center as one endpoint and a point on the circle as the other endpoint. The distance from the center to a point on the circle. C the plural of radius is radii

definitions Congruent Circles Diameter Circles with the same radius A line segment with endpoints on the circle that contains the center of the circle. The distance across a circle through the center. 2r = d

2r = d definitions Congruent Circles Diameter Circles with the same radius Diameter A line segment with endpoints on the circle that contains the center of the circle. The distance across a circle through the center. 2r = d P C R 2r = d

definitions Chord Secant Tangent A segment whose endpoints are on the circle. A diameter is a “specialized” chord. Secant A line that intersects a circle in two points. Tangent A line that intersects a circle in exactly one point. The circle and line must lie in the same plane.

definitions T U Chord B Secant Tangent C A F G A segment whose endpoints are on the circle. A diameter is a “specialized” chord. Secant A line that intersects a circle in two points. Tangent A line that intersects a circle in exactly one point. The circle and line must lie in the same plane. B C A F G

Identify each line or segment secant A chord Q radius tangent diameter radius P C X radius chord B Identify each line or segment

definition Tangent Circles externally tangent circles Coplanar circles that intersect in exactly one point. externally tangent circles

definition Tangent Circles internally tangent circles Coplanar circles that intersect in exactly one point. internally tangent circles

definition Concentric Circles Coplanar circles with a common center.

definition Common Tangent A line or segment that is tangent to two coplanar circles Common Internal Tangent crosses between the circles Common External Tangent stays along the edges of the circles

definition Common Internal Tangent Common Tangent A line or segment that is tangent to two coplanar circles Common Internal Tangent crosses between the circles Common External Tangent stays along the edges of the circles Common External Tangent

example Is the common tangent internal or external? D C T external

example tangent diameter chord radius secant Describe each segment H E F chord I A radius secant D

definitions Interior of a Circle Exterior of a Circle The set of points inside the circle Exterior of a Circle The set of points outside the circle. interior exterior

theorem t Circle Tangent-Radius Perpendicular If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. P t C

theorem t Circle Tangent-Radius Perpendicular Converse If a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is tangent to the circle. P t C

example B 11 43 C 45 A

example C r r 16 A 24 B

theorem Congruent Tangents If two segments from the same exterior point are tangent to a circle, then the segments are congruent. P S C Q

example D x2 – 4 C A 21 B

example D 15 C x 36 z y 15 A 36 B

example D 15 C 36 39 z 15 A 36 B

example What are the coordinates of each center? y x B x What is the radius of each circle?

example Describe the intersection of the two circles. y A B x

example Describe the common tangents of the circles. y A B x

example What are the coordinates of each center? y What is the radius of each circle? A B Describe any common tangents. x

C A B D Statement Reason