10.1 Use Properties of TangentsGoal Use properties of a tangent to a circle.

VOCABULARY

Circle - A circle is the set of all points in a plane that are equidistant from a given point.

Radius - A segment from the center of a circle to any point on the circle is a radius.

Chord - A segment whose endpoints are on a circle.

Diameter - A diameter is a chord that contains the center of the circle.

Secant - A line that intersects a circle in two points.

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

Example - Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangentof C.

a.

b.

c.

Example - Use the diagram to find the given lengths.

a.Radius of A

b.Diameter of A

c.Radius of B

d.Diameter of B

CIRCLES CAN INTERSECT IN:

2 points / 1 point / NO points

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

Example

Identify the point(s), line(s), or segment(s) that is (are) a:

a. Point of tangency b. Common external tangent

c. Common internal tangent d. Center

e. Radius f. Diameter

THEOREM: In a plane, a line is tangent to a circle if and only if the line is ______

to a radius of the circle at its endpoint on the circle.

Example: In the diagram, is a radius of R. Is tangent to R?Example: In the diagram,

(Is it perpendicular?)Bis a point of tangency.

Find the radius rof C.

THEOREM: Tangent segments from a

common external point are______

Example - is tangent to ⊙L at K and is tangent to ⊙L at M.

Find the value of x.

Example - The points K and M are points of tangency. Find the value(s) of x.

Algebra Skills Review

  1. Multiply BinomialsDistribute, Distribute, Distribute!!!

Example: Example: Example:

  1. Solving QuadraticsStandard Form:

Example:

Example: Example: Example:

In the diagram, assume that segments are tangents if they appear to be. Find the value(s) of the variable.

Example:In the diagram, Kis a point of tangency. Find the radius rof L

Example - Swimming Pool: You are standing 36 feet from a circular swimming pool. The distance from you to a point of tangency on the pool is 48 feet as shown. What is the radius of the swimming pool?

GSP ACTIVITY

  1. Use the compass tool to draw a circle with center A. Create a point on the circle labeled C and create a radius by selecting A and C and choosing Construct -> Segment.
  1. Create a tangent line to the circle at point C by selecting your radius and point C and choosing Construct -> Perpendicular Line.
  1. Now create another radius to another point D on the circle and another line that is tangent to the circle at point D by selecting segment AD and point D and choosing Construct -> Perpendicular Line.
  1. Create a point of intersection E between your 2 tangent lines by selecting the intersection. (Note: if your intersection is not visible on the screen move point D until it is).
  1. Measure ED by selecting E and D and choosing Measure -> Distance. Repeat for EC.
  1. Now measure one of your radii by selecting A and C (or D) and choosing Measure -> Distance.
  1. From the measure menu, choose Measure -> Calculate… From the dropdown selection button choose sqrt. Select your radius measure, ^ 2, + your (AC or AD) measure, ^ 2, and hit OK. What did you just calculate? .
  1. Confirm this result by selecting A and E and choosing Measure -> Distance.