Chapter 10

A circle is the set of points in a plane that are equal distance, the radius (r), from a given point, the center, which is also in the plane. Twice the radius of a circle is called the diameter of the circle. r

The center of the circle is point A. A circle is named by its center The center of the circle is point A. A circle is named by its center. This circle is called circle A Another definition of a radius of a circle is a segment whose endpoints are the center of the circle and a point on the circle. In circle A segment AB is a radius. A B

Chord of a Circle A chord of a circle is a segment whose endpoints are on the circle. C Segment CB is a chord of circle A. A B

A diameter of a circle is a segment that passes through the center of the circle and whose endpoints are on the circle. Segment DB is a diameter of circle A. D A B

Tangent of a Circle A tangent of a circle is a line that intersects the circle in exactly one point. A E Line BE is a tangent of circle A. B Point B is the point of tangency.

Secant of a Circle A secant of a circle is a line, a ray, or a segment that contains a chord of a circle. A Line BC is a secant of circle A. C B

Example 1 Tell whether the line or segment is best described as a radius, chord, diameter, secant, or tangent of circle C.

Example 2 Use the diagram to find the given lengths

Coplanar Circles Two coplanar circles can intersect in two points, one point, or no points.

Coplanar circles that intersect in one point are called tangent circles. Internally Tangent Circles Externally Tangent Circles

Coplanar circles that have a common center are called concentric circles.

Common Tangents A line, ray, or segment that is tangent to two coplanar circles is called a common tangent.

Example 3 Tell how many common tangents the circles have and draw them.

2 common tangents

4 common tangents

3 common tangents

1st Tangent Theorem A line, a ray, or a segment is tangent to a circle if and only if it is in the same plane as the circle and is perpendicular to a radius of the circle at the point of intersection.

Line BE is a tangent of circle A if and only if it is perpendicular to radius AB at point B.

Example 4 In the diagram, segment AB is a radius of circle A. Is segment BC tangent to circle A? Explain. Segment BC is tangent to circle A if segment BC  radius AB at pt. B. Therefore Δ ABC is not a right Δ and segment BC is not perpendicular to radius AB.

Example 5 In the diagram, S is a point of tangency. Find the radius r of circle T.

2nd Tangent Theorem Tangent segments from a common external point are congruent.

A B C

Example 6 In circle C, segment DA is tangent at A and segment DB is tangent at B. Find x.