Circle Theorems: Congruent Arcs, Chords, and Segments

1. M—06/01/09—HW #75: Pg 654-655: 1-20, skip 9, 16. Pg 821-822: 16—38 even, 39—45 odd 2) C 4) B 6) C 8) A 10) 5 12) EF = 40, FG = 50, EH = 240 14) (3\2, 2); (5\2) 18) BC, DC 20) BH = 106, HD = 74 16) 35 18) 125 20) 90 22) 90 24) 65 26) x = 80, y = 100 28) 90 30) 40 32) 46 34) 3 36) 25 38) 6

2. Chapter 10 Review

3. Angles – Middle Same • Inside – Add, div 2 • On – Half • Outside, subtract, div 2 • Segments – Tan congruent • Inside, part part = part part • Outside, part whole = part whole pr • Part whole = whole squared

4. A O P Theorem 10.3 Tangents to a circle from a point are congruent. B

5. E D F A B C Also state if they are major or minor arcs

6. Theorem 10.4 In the same circle or in congruent circles: 1) Congruent arcs have congruent chords. 2) Congruent chords have congruent arcs. W O R G N

7. Theorem 10.5 A diameter that is perpendicular to a chord bisects the chord and its arc. B R N Theorem 10.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. E A

8. Theorem 10.7 In the same circle or in congruent circles. 1) Chords equally distant from the center (or centers) are congruent. 2) Congruent chords are equally distant from the center (or centers) Remember, shortest distance means perpendicular. B R N E A

9. z 5 4 y x

10. Def: An inscribed angle is an angle whose vertex is on the circle and sides are chords in the circle. Theorem: Measure of inscribed angle is half the measure of the intercepted arc.

11. Look at the pictures, what can you conclude? Two inscribed angles intercept same arc, then they are congruent Angle inscribed in semi circle is 90 degrees Proof: 180\2 If quadrilateral is inscribed in circle, opposite angles supplementary

15. Theorem 10-12 Measure of angle formed by chord and tangent is half the intercepted arc Chord tan half arc

16. Thrm: Measure of angles formed by two chords that intersect inside the circle is equal to half the sum of the measures of the intercepted arcs.

17. The measure of an angle formed by two secants, two tangents, or a secant and tangent drawn from a point OUTSIDE the circle is equal to half the difference of the measures of the intercepted arcs.

18. 80, inscribed angle 25, .5(80-30) 50, .5(100-z)=25

19. 80, .5(x+160)=120 40, .5(160-80)=y

20. 10.15 Two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. 150=10a 15=a

21. Theorem 10.16: When two secants are drawn from an external point, the product of one secant with external segment is equal to the product of the other secant and external segment. Theorem 10.17: When a secant and tangent are drawn to a circle from an external point, the product of the secant and its external segment is equal to the tangent square. External Segment ab=c2 ab=cd

22. D A E C F B 8 18

23. A 4 or 12 C E D B

24. Graphing a circle, give radius, center (x – 3)2 + (y + 1)2 = 25 (h,k) is the center (3,–1) • Plot center • Find radius • Plot points • Connect r is the radius

25. Write equation given center and radius • Center • Radius • Center • Radius (2,–4) (–1,2) (x – (– 1))2 + (y – 2)2 = 32 (x – 2)2 + (y –(– 4))2 =( )2 (x – 2)2 + (y + 4)2 = 21 (x + 1)2 + (y – 2)2 = 9

26. M—06/01/09—HW #75: Pg 654-655: 1-20, skip 9, 16. Pg 821-822: 16—38 even, 39—45 odd