29. Think About a Plan In the diagram, the circles are concentric. What is a formula you could use to find the value of c in terms of a and b?
    • How can you use the inscribed angle to find the value of c?
    • What is the relationship of the inscribed angle to a and b?

    

30. Δ P Q R cap delta p q r  is inscribed in a circle with m ∠ P = 70 , m ∠ Q = 50 , m angle . p equals 70 comma . m angle . q equals 50 comma  and m ∠ R = 60 . m angle . r equals 60 .  What are the measures of P Q ⌢ , Q R ⌢ , modified p q with frown above , comma . modified q r with frown above , comma  and P R ⌢ ? modified p r with frown above , question mark

31. Reasoning Use the diagram below. If you know the values of x and y, how can you find the measure of each numbered angle?

    

Algebra Find the values of x and y using the given chord, secant, and tangent lengths. If your answer is not a whole number, round it to the nearest tenth.

32. 
33. 
    Image Long Description
34. 

35. Proof Prove Theorem 12-14 as it applies to two secants that intersect outside a circle.

    Given: ⊙ O circle dot o  with secants C A ¯ c eh bar  and C E ¯ c e bar

    Prove: m ∠ A C E = 1 2 ( m A E ⌢ − m B D ⌢ ) ( m A E ⌢ − m B D ⌢ ) m angle eh c e equals , 1 half . open . m , modified eh e with frown above , minus m , modified b d with frown above . close . open m , modified eh e with frown above . minus . m , modified b d with frown above , close

    

36. Proof Prove the other two cases of Theorem 12-14. (See Exercise 35.)

For Exercises 37 and 38, write proofs that use similar triangles.

37. Proof Prove Theorem 12-15, Case II.
38. Proof Prove Theorem 12-15, Case III.
39. The diagram below shows a unit circle, a circle with radius 1.
    1. What triangle is similar to Δ A B E ? cap delta eh b e , question mark
    2. Describe the connection between the ratio for the tangent of ∠ A angle eh  and the segment that is tangent to ⊙ A . circle dot eh .

    3. The secant ratio is hypotenuse length of leg adjacent to an angle . fraction hypotenuse , over lengthoflegadjacenttoanangle end fraction . .  Describe the connection between the ratio for the secant of ∠ A angle eh  and the segment that is the secant in the unit circle.

       

C Challenge

For Exercises 40 and 41, use the diagram below. Prove each statement.

40. Proof m ∠ 1 + m P Q ⌢ = 180 m angle , 1 plus . m , modified p q with frown above . equals 180

41. Proof m ∠ 1 + m ∠ 2 = m Q R ⌢ m angle , 1 plus , m angle , 2 equals . m , modified q r with frown above

    

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