1. Chapter 11 Circles

2. Section 11 – 1 Tangent Lines
Objectives:
To use the relationship between a radius and a tangent
To use the relationship between two tangents from one point

3. A Tangent to a Circle: Point of Tangency:
A line, ray or segment in the plane of a circle that intersects the circle in exactly one point.
A Tangent
Point of Tangency
Point of Tangency:
The point where a circle and a tangent intersect

4. Theorem 11 – 1
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

5. Example 1 Finding Angle Measures
A) 𝑀𝐿 and 𝑀𝑁 are tangent to ⨀ O. Find the value of x.

6. Example 1 Finding Angle Measures
B) 𝐸𝐷 is tangent to ⨀ O. Find the value of x.

7. Example 1 Finding Angle Measures
C) 𝐵𝐴 is tangent to ⨀ C at point A. Find the value of x.

8. Example 2 Real-World Connection
A) A dirt bike chain fits tightly around two gears. The chain and gears form a figure like the one at the right. Find the distance between the centers of the gears.

9. Example 2 Real-World Connection
B) A belt fits tightly around two circular pulleys, as shown at the right. Find the distance between the centers of the pulleys.

10. Example 2 Real-World Connection
C) A belt fits tightly around two circular pulleys, as shown at the right. Find the distance between the centers of the pulleys.

11. Theorem 11 – 2 (Converse of 11 – 1)
If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

12. Example 3 Finding a Tangent
A) Is 𝑀𝐿 tangent to ⨀ N at L? Explain.

13. Example 3 Finding a Tangent
B) If NL = 4, LM = 7 and NM = 8, is 𝑀𝐿 tangent to ⨀ N at L? Explain.

14. Example 3 Finding a Tangent
C) ⨀ O has radius 5. Point P is outside ⨀ O such that PO = 12, and Point A is on ⨀ O such that PA = 13. Is 𝑃𝐴 tangent to ⨀ O at A? Explain.

15. HOMEWORK: Textbook Page 586; #1 – 4, 6, 10 – 12

17. Section 11 – 1 Continued… Objectives:
To use the relationship between two tangents from one point

18. Triangle Inscribed In a Circle
Triangle Circumscribed about a Circle

19. Theorem 11 – 3
The two segments tangent to a circle from a point outside the circle are congruent.

20. Example 5 Circles Inscribed in Polygons
A) ⊙ O is inscribed in ∆ ABC. Find the perimeter of ∆ABC.

21. Example 5 Circles Inscribed in Polygons
B) ⊙ O is inscribed in ∆ PQR. ∆PQR has a perimeter of 88 cm. Find QY.

22. Example 5 Circles Inscribed in Polygons
C) ⊙ C is inscribed in quadrilateral XYZW. Find the perimeter of XYZW.

23. Additional Examples
Assume the lines that appear to be tangent are tangent. O is the center of each circle. Find the value of x.

24. Homework
11 – 1 Ditto; # 1 – 15 Odds