1. Use Properties of Tangents
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
1. What measure is needed to find the circumference or area of a circle?
ANSWER
radius or diameter
2. Find the radius of a circle with diameter 8 centimeters.
ANSWER
4 cm
3. A right triangle has legs with lengths 5 inches and 12 inches. Find the length of the hypotenuse.
ANSWER
13 in.

3. Warm-Up
4. Solve 6x + 15 = 33.
ANSWER 3
5. Solve (x + 18)2 = x
ANSWER 7

4. Example 1
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. AC a. b. AB
SOLUTION
is a radius because C is the center and A is a point on the circle. AC a. b. AB
is a diameter because it is a chord that contains the center C.

5. Example 1
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. DE c. AE d.
SOLUTION c. DE
is a tangent ray because it is contained in a line that intersects the circle at only one point. d. AE
is a secant because it is a line that intersects the circle in two points.

6. Guided Practice
1. In Example 1, what word best describes AG? CB?
ANSWER
chord; radius
2. In Example 1, name a tangent and a tangent segment.
ANSWER
DE, DB

7. Example 2
Use the diagram to find the given lengths. a.
Radius of A b.
Diameter of A
Radius of B c.
Diameter of B d.
SOLUTION a.
The radius of A is 3 units. b.
The diameter of A is 6 units. c.
The radius of B is 2 units. d.
The diameter of B is 4 units.

8. Guided Practice
3. Use the diagram in Example 2 to find the radius and diameter of C and D.
ANSWER
3; 6; 2; 4

9. Example 3
Tell how many common tangents the circles have and draw them. a.
SOLUTION a.
4 common tangents

10. Example 3
Tell how many common tangents the circles have and draw them. b.
SOLUTION
3 common tangents b.

11. Example 3
Tell how many common tangents the circles have and draw them. c.
SOLUTION c.
2 common tangents

12. Guided Practice
Tell how many common tangents the circles have and draw them. 4.
2 common tangents
ANSWER

13. Guided Practice
Tell how many common tangents the circles have and draw them. 5.
1 common tangent
ANSWER

14. Guided Practice
Tell how many common tangents the circles have and draw them. 6.
no common tangents
ANSWER

15. Example 4
In the diagram, PT is a radius of P. Is ST tangent to P ?
SOLUTION
Use the Converse of the Pythagorean Theorem. Because = 372, PST is a right triangle and ST PT . So, ST is perpendicular to a radius of P at its endpoint on P. By Theorem 10.1, ST is tangent to P.

16. In the diagram, B is a point of tangency. Find the radius r of C.
Example 5
In the diagram, B is a point of tangency. Find the radius r of C.
SOLUTION
You know from Theorem 10.1 that AB BC , so ABC is a right triangle. You can use the Pythagorean Theorem.
AC2 = BC2 + AB2
Pythagorean Theorem
(r + 50)2 = r
Substitute.
r r = r
Multiply.
100r = 3900
Subtract from each side.
r = 39 ft
Divide each side by 100.

17. Example 6
RS is tangent to C at S and RT is tangent to C at T. Find the value of x.
SOLUTION
RS = RT
Tangent segments from the same point are .
28 = 3x + 4
Substitute.
8 = x
Solve for x.

18. Guided Practice
7. Is DE tangent to C?
ANSWER
Yes

19. Guided Practice
8. ST is tangent to Q.Find the value of r.
ANSWER 7

20. Guided Practice
9. Find the value(s) of x.
ANSWER ±3

21. Lesson Quiz 1.
Give the name that best describes the figure . a. CD b. AB
ANSWER
secant
ANSWER
tangent c. FD d. EP
ANSWER
chord
ANSWER
radius

22. Lesson Quiz 2.
Tell how many common tangents the circles have .
ANSWER
One tangent; it is a vertical line through the point of tangency.

23. Lesson Quiz 3.
Is AB tangent to C? Explain. .
ANSWER
Yes; = 1156 = 34 so AB AC, and a line to a radius at its endpoint is tangent to the circle. 2

24. Lesson Quiz 4.
Find x.
ANSWER 12