Lesson 9-2 Tangents (page 333) Essential Question How can relationships in a circle allow you to solve problems involving segments, angles, and arcs?

Tangents

Theorem 9-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. O m T

This theorem is proven with an indirect proof This theorem is proven with an indirect proof. Also, this is why the tangent term is used in trigonometry. If O is a unit circle, the radius = 1, then … O 1 m T Z

Tangents to a circle from a point are congruent . Corollary Tangents to a circle from a point are congruent . A P O B

The tangents to a circle from a point are really the tangent segments. B Tangent segments to a circle from a point are congruent .

To prove this corollary, prove ∆PAO ≅ ∆PBO by the HL Theorem. Then PA = PB by CPCTC.

Theorem 9-2 If a line in the plane of a circle is perpendicular to the radius at its outer endpoint, then the line is tangent to the circle. Q l R

This is the converse of Theorem 9-1 and it also is proven with an indirect proof. Q l R

example: RS is tangent to P at R. If PR = 5 and RS = 12, find PS.

CIRCUMSCRIBED POLYGON … a polygon is circumscribed about a circle if each of its sides is tangent to the circle.

INSCRIBED CIRCLE … a circle is inscribed in a polygon if each of the sides of the polygon is tangent to the circle.

CIRCUMSCRIBED POLYGON and INSCRIBED CIRCLE are the same thing!

COMMON TANGENT … a line that is tangent to each of two coplanar circles.

COMMON INTERNAL TANGENT … intersects the segment joining the centers of the circles.

Here is another COMMON TANGENT.

COMMON EXTERNAL TANGENT … does not intersect the segment joining the centers of the circles.

Are there anymore COMMON TANGENTS?

TANGENT CIRCLES … two coplanar circles that are tangent to the same line at the same point. #1 Externally Tangent Circles

TANGENT CIRCLES … two coplanar circles that are tangent to the same line at the same point. #2 Internally Tangent Circles

Classroom Exercises on page 335 Classroom Assignment Classroom Exercises on page 335 1 to 3 all numbers How can relationships in a circle allow you to solve problems involving segments, angles, and arcs?

Assignment Written Exercises on pages 335 to 337 RECOMMENDED: 1 to 7 odd numbers, 11 REQUIRED: 9, 16, 17, 18, 20, 21 How can relationships in a circle allow you to solve problems involving segments, angles, and arcs?