27. In the diagram below, the endpoints of the chord are the points where the line x = 2 x equals 2  intersects the circle x 2 + y 2 = 25 . x squared , plus , y squared . equals 25 .  What is the length of the chord? Round your answer to the nearest tenth.

    
28. Construction Use a circular object such as a can or a saucer to draw a circle. Construct the center of the circle.
29. Writing Theorems 12-4 and 12-5 both begin with the phrase, “within a circle or in congruent circles.” Explain why the word congruent is essential for both theorems.

Find m A B ⌢ . m , modified eh b with frown above , .  (Hint: You will need to use trigonometry in Exercise 32.)

30. 
31. 
32. 

33. Proof Prove Theorem 12-10.

    Given: l is the ⊥ up tack  bisector of W Y ¯ . w y bar , .

    Prove: l contains the center of ⊙ X . circle dot x .

    

34. Proof Given: ⊙ A circle dot eh  with C E ¯ ⊥ B D ¯ c e bar , up tack , b d bar

    Prove: B C ⌢ ≅ D C ⌢ modified b c with frown above , approximately equal to , modified d c with frown above

    

C Challenge

Proof Prove each of the following.

35. Converse of Theorem 12-4: Within a circle or in congruent circles, congruent arcs have congruent central angles.
36. Converse of Theorem 12-5: Within a circle or in congruent circles, congruent chords have congruent central angles.
37. Converse of Theorem 12-6: Within a circle or in congruent circles, congruent arcs have congruent chords.
38. Converse of Theorem 12-7: Within a circle or congruent circles, congruent chords are equidistant from the center (or centers).
39. Proof If two circles are concentric and a chord of the larger circle is tangent to the smaller circle, prove that the point of tangency is the midpoint of the chord.

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