1. Circles, II
Chords
Arcs

2. Definition – Arc
A central angle separates the circle into two parts, each of which is an arc. The measure of each arc is related to the measure of its central angle.

3. Arcs

4. Arc Addition Postulate
The measure of an arc formed by two adjacent (neighboring) arcs is the sum of the measures of the two arcs.

5. Example 1 – Measures of Arcs
Compute the measurement of arc BE

6. Example 1 – Measures of Arcs
Compute the measurement of arc BE

7. Example 1 – Measures of Arcs
Compute the measurement of arc BE

8. Example 1 – Measures of Arcs
Compute the measurement of arc BE

9. Example 2 – Measures of Arcs
Compute the measurement of arc CE

10. Example 2 – Measures of Arcs
Compute the measurement of arc CE
Arc CE is a minor arc

11. Example 2 – Measures of Arcs
Compute the measurement of arc CE
Arc CE is a minor arc
𝐶𝐸 = 𝐶𝐵 + 𝐵𝐸

12. Example 2 – Measures of Arcs
Compute the measurement of arc CE
Arc CE is a minor arc
𝐶𝐸 = 𝐶𝐵 + 𝐵𝐸
𝐶𝐸 =90+50=140

13. Example 3 – Measures of Arcs
Compute the measurement of arc ACE

14. Example 3 – Measures of Arcs
Compute the measurement of arc ACE
Arc ACE is a major arc

15. Example 3 – Measures of Arcs
Compute the measurement of arc ACE
Arc ACE is a major arc
𝐴𝐶𝐸 = 𝐴𝐷 + 𝐷𝐵𝐸

16. Example 3 – Measures of Arcs
Compute the measurement of arc ACE
Arc ACE is a major arc
𝐴𝐶𝐸 = 𝐴𝐷 + 𝐷𝐵𝐸
𝐴𝐶𝐸 =50+180=230

17. CW – Arcs

18. Arcs and Chords – 2 Theorems
In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

19. Arcs and Chords – 2 Theorems
In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

20. Arcs and Chords – 2 Theorems
In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
𝐴𝐵 =80

21. Arcs and Chords – 2 Theorems
In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
𝐴𝐵 =80
𝐴𝐵 =80= 𝐶𝐷

22. Arcs and Chords – 2 Theorems
In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects (cut in half) the chord and its arc.

23. Arcs and Chords – 2 Theorems
In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects (cut in half) the chord and its arc.

24. Example 1
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

25. Example 1
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

26. Example 1
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

27. Example 1
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

28. Example 2
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.
Compute length of segment OX.

29. Example 2
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.
Compute length of segment OX.
Draw segment CO, CO =

30. Example 2
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.
Compute length of segment OX.
Draw segment CO, CO = 13

31. Example 2
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.
Compute length of segment OX.
Draw segment CO, CO = 13
CX = 12, by the second theorem.

32. Example 2
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.
Compute length of segment OX.
Draw segment CO, CO = 13
CX = 12, by the second theorem.

33. Example 2
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.
Compute length of segment OX.
Draw segment CO, CO = 13
CX = 12, by the second theorem.

34. Example 2
Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.
Compute length of segment OX.
Draw segment CO, CO = 13
CX = 12, by the second theorem.

35. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿

36. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

37. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

38. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

39. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

40. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

41. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

42. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

43. Example 3
Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long.
Compute 𝐽𝐿
Draw 𝑊𝐾

44. CW 2, HW