Chapter 10 Circles

1. Identify segments and lines related to circles Use properties of tangents to circles. Chapter 10Circles Section 10.1 Tangents

2. Identify segments and lines related to circles Use properties of tangents to circles. Lesson 10-1 Contents Key Concepts Example 1Identifying Parts of a Circle Example 2Finding length of Radii Example 3Line Tangent to a Circle Example 4Determining Tangency Example 5Finding Lengths of Tangents

3. Circles Definition C Definition Circle A circle is the set of all points in a plane that are equidistant from a given point, called the center of the circle. Circle C The distance from the center to a point on the circle is called the radius

4. Circles Any segment with endpoints being the center and any point on the circle is called the Radius Any Segment with endpoints on the circle are called chords A chord that goes through the center is called a diameter A diameter is composed of two radii

5. Circles Secant: A line in the same plane that intersects a circle twice Tangent: A line in the same plane that intersects a circle exactly once Point of Tangency

6. Circles

7. Identify segments and lines related to circles Answer: The circle has its center at E, so it is named circle E, or . Example 1-1a Name the circle.

8. Identify segments and lines related to circles Answer: Four radii are shown: . Example 1-1b Name the radius of the circle.

9. Identify segments and lines related to circles Answer: Four chords are shown: . Example 1-1c Name a chord of the circle.

10. Identify segments and lines related to circles Answer: are the only chords that go through the center. So, are diameters. Example 1-1d Name a diameter of the circle.

11. Identify segments and lines related to circles a. Name the circle.b. Name a radius of the circle. c. Name a chord of the circle. d. Name a diameter of the circle. Answer: Answer: Answer: Answer: Example 1-1e

12. Identify segments and lines related to circles Circle R has diameters and . If ST18, find RS. Example 1-2a Formula for radius Substitute and simplify. Answer: 9

13. Identify segments and lines related to circles Circle R has diameters . If RM24, find QM. Example 1-2b Formula for diameter Substitute and simplify. Answer: 48

14. Identify segments and lines related to circles Circle R has diameters . If RN2, find RP. Example 1-2c Since all radii are congruent, RN=RP. Answer: So, RP=2.

15. Identify segments and lines related to circles Circle M has diameters a. If BG=25, find MG. b. If DM=29, find DN. c. If MF=8.5, find MG. Example 1-2d Answer: 12½ Answer: 58 Answer: 8½

16. Use properties of tangents to circles. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

17. ALGEBRA is tangent to at point R. Find y. Because the radius is perpendicular to the tangent at the point of tangency, . This makes a right angle and  a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y. Use properties of tangents to circles. Example 5-1a

18. Use properties of tangents to circles. Answer: Thus, y is twice . Example 5-1b Pythagorean Theorem Simplify. Subtract 256 from each side. Take the square root of each side. Because y is the length of the diameter, ignore the negative result.

19. is a tangent to at point D. Find a. Use properties of tangents to circles. Example 5-1c Answer: 15

20. Use properties of tangents to circles. If a line is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

21. Determine whether is tangent to Use properties of tangents to circles. Example 5-2a First determine whether ABCis a right triangle by using the converse of the Pythagorean Theorem.

22. Answer: So, is not tangent to . Use properties of tangents to circles. Example 5-2b Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem did not prove true in this case, ABC is not a right triangle.

23. Determine whether is tangent to Use properties of tangents to circles. Example 5-2c First determine whether EWDis a right triangle by using the converse of the Pythagorean Theorem.

24. Answer: Thus, making a tangent to Use properties of tangents to circles. Example 5-2d Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle.

25. a. Determine whether is tangent to Use properties of tangents to circles. Example 5-2e Answer: yes

26. b. Determine whether is tangent to Use properties of tangents to circles. Example 5-2f Answer: no

27. Use properties of tangents to circles. If two segments from the same exterior point are tangent to a circle, then they are congruent.

28. are drawn from the same exterior point and are tangent to so are drawn from the same exterior point and are tangent to Use properties of tangents to circles. Example 5-3a ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent.

29. Use properties of tangents to circles. Example 5-3b Definition of congruent segments Substitution. Use the value of y to find x. Definition of congruent segments Substitution Simplify. Subtract 14 from each side. Answer: 1

30. Use properties of tangents to circles. Example 5-3d ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent. Answer: –6

31. Triangle HJK is circumscribed about Find the perimeter of HJK if Use properties of tangents to circles. Example 5-4a

32. We are given that Use properties of tangents to circles. Example 5-4b Use Theorem 10.3 to determine the equal measures. Definition of perimeter Substitution Answer: The perimeter of HJKis 158 units.

33. Triangle NOT is circumscribed about Find the perimeter of NOT if Use properties of tangents to circles. Example 5-4c Answer: 172 units

34. HW # 34Pg 599-600 9-35