properties of tangents

Properties of TangentsObjectives:

1. To define and use circle terminology2. To use properties of tangents to a circle

Tangent

A tangent is a line that intersects a circle at exactly one point.

– The point of intersection is called the point of tangency

Tangent

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

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

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

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

Common Tangents, IICommon 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

Example 5, RevisitedDetermine whether the common tangents are internal or external.

Tangent Line TheoremIn 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.

Example 6The 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).

6

4

2

-2

5

Example 7•  

Example 8•  

Example 9•  

Example 10•  

Congruent Tangents Theorem

Tangent segments from a common external point are congruent.

Example 11•  

Challenge ProblemA 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.