Geometry 9.2 Tangents

Tangent Definition A line in the plane of a circle that intersects the circle in exactly one point. tangent

Theorem If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. ●

Converse of the previous Theorem If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle. ●

Corollary Tangents to a circle from a point are congruent. ● ●

Exercises 1. If OC = 15 and OB = 9, then BC = 12 3-4-5 Triple 15 9 2. If OC = 3√6 and BC = 6, then OB = 3√2 3√6 x²+ 6² = (3√6)² x x²+ 36 = 54 6 x² = 18 x = √18 = 3√2

Exercises 6. If m<COB = 60 and CB = 6√3, then AB = 12 30-60-90 Rt. ∆ 60˚ 6√3 7. If m<BCD = 70, then m<CBD = m<____ = CDB 55 Iso. ∆ 70˚

Answers to Exercises O B C D A 10 6√2 8 8. 25

Common Tangents A line that is tangent to each of two coplanar circles. common tangent ● ● common tangent

Internal vs. External Tangents A common external tangent does not. common external tangent ● ● A common internal tangent intersects the segment joining the centers of the circle. common internal tangent

Tangent Circles Two coplanar circles that are tangent to the same line at the same point. There are two types: internally tangent circles externally tangent circles

Circumscribed Polygons A polygon is circumscribed about a circle if each of its sides is tangent to the circle. A circle is inscribed in a polygon if each of the sides of the polygon is tangent to the circle.

Exercises Tell whether the circles are externally tangent, internally tangent, or not tangent. Tell whether the line is a common external tangent, a common internal tangent, or neither. 9. 10. 11. not tangent com. ext. tangent a. int. tangent b. com. ext. tangent a. not tangent b. neither

Exercises Tell whether the circles are externally tangent, internally tangent, or not tangent. Tell whether the line is a common external tangent, a common internal tangent, or neither. 12. 13. 14. ext tangent com. int. tangent a. not tangent b. com. int. tangent a. not tangent b. com ext. tangent

Exercises Tell whether the circles are externally tangent, internally tangent, or not tangent. Tell whether the line is a common external tangent, a common internal tangent, or neither. 15. Find AC 16. Find SP and RQ. 17.Find PT and QS 3 2 1 C B A P Q R S 15 17 24 23 P 13 9 S 24 23 R 13 9 T 15 17 Q AC = 8 SP = 39 RQ = 40 PT = 22 QS = 22

Homework pg. 335 WE #1-6, 8-10, 14-18