10.1 Use Properties of Tangents Goal  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 tangent of C.
a. BC

b. EA

c. DE
Example - Use the diagram to find the given lengths.
a.
b.
c.
d.
Radius of A
Diameter of A
Radius of B
Diameter of B
CIRCLES CAN INTERSECT IN:
2 points
1 point
NO points
Definition:
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
c. Common internal tangent
e. Radius
b. Common external tangent
d. Center
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, RS is a radius of R. Is ST tangent to R?
Example: In the diagram,
(Is it perpendicular?)
B is a point of tangency.
Find the radius r of C.
THEOREM: Tangent segments from a
common external point are__________
Example - JK is tangent to ⊙L at K and JM 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
I.
Multiply Binomials
Example: 𝟑(𝒙 + 𝟑)
II.
Solving Quadratics
Distribute, Distribute, Distribute!!!
Example: (𝒙 + 𝟑)(𝒙 − 𝟐)
Standard Form:
Example:
Example: 𝒙𝟐 + 𝟐𝒙 − 𝟖 = 𝟎
Example: (𝒓 + 𝟐)𝟐
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
𝑥 2 + 2𝑥 − 8 = 0
Example: 𝒙𝟐 + −𝟔 = 𝒙
Example: 𝟑𝒙𝟐 = 𝟕𝟓
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝐹𝑜𝑟𝑚𝑢𝑙𝑎: 𝑥 =
(always works!)
2𝑎
In the diagram, assume that segments are tangents if they appear to be. Find the value(s) of the variable.
Example: In the diagram, K is a point of tangency. Find the radius r of 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.
2. Create a tangent line to the circle at point C by selecting your radius and point C and choosing
Construct -> Perpendicular Line.
3. 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.
4. 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).
5. Measure ED by selecting E and D and choosing Measure -> Distance. Repeat for EC.
6. Now measure one of your radii by selecting A and C (or D) and choosing Measure -> Distance.
7. 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? √𝑟𝑎𝑑𝑖𝑢𝑠 2 + 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 2 .
8. Confirm this result by selecting A and E and choosing Measure -> Distance.