1. Geometry
9.2
Tangents

2. Tangent Definition
A line in the plane of a circle that intersects the circle in exactly one point.
tangent

3. Theorem
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. ●

4. Converse of the previous Theorem
If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle. ●

5. Corollary
Tangents to a circle from a point are congruent. ● ●

6. Exercises 1. If OC = 15 and OB = 9, then BC = 12
3-4-5 Triple 15 9
2. If OC = 3√6 and BC = 6, then OB =
3√2
3√6
x²+ 6² = (3√6)² x
x²+ 36 = 54 6
x² = 18
x = √18 = 3√2

7. Exercises 6. If m<COB = 60 and CB = 6√3, then AB = 12
Rt. ∆
60˚
6√3
7. If m<BCD = 70, then m<CBD = m<____ =
CDB 55
Iso. ∆
70˚

8. Answers to Exercises O B C D A 10
6√2 8
8. 25

9. Common Tangents
A line that is tangent to each of two coplanar circles.
common tangent ● ●
common tangent

10. Internal vs. External Tangents
A common external tangent does not.
common external tangent ● ●
A common internal tangent intersects the segment joining the centers of the circle.
common internal tangent

11. Tangent Circles
Two coplanar circles that are tangent to the same line at the same point. There are two types:
internally tangent circles
externally tangent circles

12. Circumscribed Polygons
A polygon is circumscribed about a circle if each of its sides is tangent to the circle.
A circle is inscribed in a polygon if each of the sides of the polygon is tangent to the circle.

13. Exercises
Tell whether the circles are externally tangent, internally tangent, or not tangent.
Tell whether the line is a common external tangent, a common internal tangent, or neither. 9.
10.
11.
not tangent
com. ext. tangent
a. int. tangent
b. com. ext. tangent
a. not tangent
b. neither

14. Exercises
Tell whether the circles are externally tangent, internally tangent, or not tangent.
Tell whether the line is a common external tangent, a common internal tangent, or neither.
12.
13.
14.
ext tangent
com. int. tangent
a. not tangent
b. com. int. tangent
a. not tangent
b. com ext. tangent

15. Exercises
Tell whether the circles are externally tangent, internally tangent, or not tangent.
Tell whether the line is a common external tangent, a common internal tangent, or neither.
15. Find AC
16. Find SP and RQ.
17.Find PT and QS 3 2 1 C B A P Q R S 15 17 24 23 P 13 9 S 24 23 R 13 9 T 15 17 Q
AC = 8
SP = 39
RQ = 40
PT = 22
QS = 22

16. Homework
pg WE #1-6, 8-10, 14-18