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CIRCLES Chapter 10

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Tangents to Circles lesson 10.1 California State Standards 7: Prove and Use theorems involving properties of circles. 21: Prove and Solve relationships among chords, secants and tangents.

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definitions 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.

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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. definitions C P the plural of radius is radii

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definitions Congruent Circles 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. 2 r = d

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definitions Congruent Circles 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

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definitions Chord 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.

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definitions Chord 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. C A B F G T U

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

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

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

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definition Concentric Circles Coplanar circles with a common center.

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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

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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 C o m m o n I n t e r n a l T a n g e n t Common External Tangent

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

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

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

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theorem 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. t C P

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C theorem 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. t P

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example C A B

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example C A B r r 16 24

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

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example A B D x 2 – 4 21 C

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example B D x y z A C 36 15

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36 example B D z A C 39 15

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

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example Describe the intersection of the two circles. x y AB

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example Describe the common tangents of the circles. x y AB

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

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Statement C A B D Reason

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