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Chapter 10

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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

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**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

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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

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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

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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.

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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

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

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Example 2 Use the diagram to find the given lengths

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

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**Coplanar circles that intersect in one point are called tangent circles.**

Internally Tangent Circles Externally Tangent Circles

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

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

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

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2 common tangents

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4 common tangents

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3 common tangents

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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.

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

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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.

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

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

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A B C

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

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