1. Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 10.3 Arcs and Chords

2. Objectives Recognize and use relationships between arcs and chords Recognize and use relationships between chords and diameters

3. Arcs and Chords The endpoints of chords also create arcs. B R A S Theorem 10.2: In a or  s, two minor arcs are  if their corresponding chords are .

4. Example 3-1a PROOF Write a proof. Prove: Given: is a semicircle.

5. Example 3-1b Proof: StatementsReasons 1. 1. Given is a semicircle. 5. Def. of arc measure 5. 2. Def. of semicircle 2. 3. In a circle, 2 chords are, corr. minor arcs are. 3. 4. Def. of arcs4.

6. Example 3-1c Answer: StatementsReasons 6. 6. Arc Addition Postulate 7. 7. Substitution 8. 8. Subtraction Property and simplify 9.9. Division Property 10. 10. Def. of arc measure 11. 11. Substitution

7. Example 3-1d PROOF Write a proof. Prove: Given:

8. Example 3-1e Proof: Statements Reasons 1. 2. 3. 4. 1. Given 2. In a circle, 2 minor arcs are, chords are. 3. Transitive Property 4. In a circle, 2 chords are, minor arcs are.

9. Arcs and Chords The chords of adjacent arcs can form a polygon. The polygon in the picture is an inscribed polygon because all of the vertices lie on the circle and the circle is a circumscribed circle because it contains all of the vertices of the polygon.

10. Example 3-2a TESSELLATIONS The rotations of a tessellation can create twelve congruent central angles. Determine whether.

11. Example 3-2b Answer: Since the measures of are equal,. Because all of the twelve central angles are congruent, the measure of each angle is Let the center of the circle be A. The measure of Then. The measure of Then.

12. Example 3-2c ADVERTISING A logo for an advertising campaign is a pentagon that has five congruent central angles. Determine whether. Answer: no

13. Diameters and Chords Diameters that are ┴ to a chord in a create special segments and arc relationships. Theorem 10.3: In a, if a diameter (or radius) is ┴ to a chord, then it bisects the chord and its arc.

14. Diameters and Chords Theorem 10.3 (continued): If JK ┴ LM, then MO  LO and arc LK  arc MK. o

15. Example 3-3a Circle W has a radius of 10 centimeters. Radius is perpendicular to chord which is 16 centimeters long. If find

16. Example 3-3b Since radius is perpendicular to chord Arc addition postulate Substitution Subtract 53 from each side. Answer: 127

17. Example 3-3c Circle W has a radius of 10 centimeters. Radius is perpendicular to chord which is 16 centimeters long. Find JL.

18. Example 3-3d A radius perpendicular to a chord bisects it. Definition of segment bisector Draw radius 

19. Example 3-3e Use the Pythagorean Theorem to find WJ. Pythagorean Theorem Simplify. Subtract 64 from each side. Take the square root of each side.

20. Example 3-3f Segment addition Subtract 6 from each side. Answer: 4

21. Example 3-3g Answer: 145 Answer: 10 Circle O has a radius of 25 units. Radius is perpendicular to chord which is 40 units long. a. If b. Find CH.

22. More About Chords Theorem 10.4: In a or in  s, two chords are  iff they are equidistant from the center.

23. Example 3-4a Chords and are equidistant from the center. If the radius of is 15 and EF = 24, find PR and RH.

24. Example 3-4b are equidistant from P, so.

25. Example 3-4c Draw to form a right triangle. Use the Pythagorean Theorem. Pythagorean Theorem Simplify. Subtract 144 from each side. Take the square root of each side. Answer:

26. Example 3-4d Answer: Chords and are equidistant from the center of If TX is 39 and XY is 15, find WZ and UV.

27. Assignment Pre-AP GeometryPre-AP Geometry Pg. 540 #11 – 36, 40, 42 Geometry:Geometry: Pg. 540 #11 – 35