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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 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.

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**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

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**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

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**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

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**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.

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**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

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**Identify each line or segment**

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

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**definition Tangent Circles externally tangent circles**

Coplanar circles that intersect in exactly one point. externally tangent circles

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**definition Tangent Circles internally 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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**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

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

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**example tangent diameter chord radius secant Describe each segment H E**

F chord I A radius secant D

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**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

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**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

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**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

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example B 11 43 C 45 A

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example C r r 16 A 24 B

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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 S C Q

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example D x2 – 4 C A 21 B

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

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example D 15 C 36 39 z 15 A 36 B

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**example What are the coordinates of each center? y x**

B x What is the radius of each circle?

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

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

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**example What are the coordinates of each center? y**

What is the radius of each circle? A B Describe any common tangents. x

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C A B D Statement Reason

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