Properties of Tangents

1. Properties of Tangents Objectives: • To define and use circle terminology • To use properties of tangents to a circle

2. Tangent A tangent is a line that intersects a circle at exactly one point. • The point of intersection is called the point of tangency

3. Tangent

4. Example 2 Explain why the wheels on a train are closer to being tangent to the rails than a car tire to the road.

5. Example 4 Draw two coplanar circles that intersect in a) two points, b) one point, c) no points and have the same center.

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

7. Example 5 Tell how many common tangents the circles have and draw them all.

8. Common Tangents, II Common tangents come in two flavors: Common Internal Tangent: Intersect the segment that joins the centers of the circles Common External Tangent: Does not intersect the segment that joins the centers of the circles

9. Example 5, Revisited Determine whether the common tangents are internal or external.

10. Tangent Line Theorem In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

11. Example 6 The center of a circle has coordinates (1, 2). The point (3, -1) lies on this circle. Find the slope of the tangent line at (3, -1).

12. Example 7

13. Example 8

14. Example 9

15. Example 10

16. Congruent Tangents Theorem Tangent segments from a common external point are congruent.

17. Example 11

18. Challenge Problem A circle has a radius of 6 inches. Two radii form a central angle of 60°. Tangent lines are drawn to the endpoints of each of the radii. How far from the center do the two tangent lines intersect? Due at end of class.