What is the measure of angle ACB in the circle above with center O?
 
(1) The radius of the circle is 5.
(2) The length of the arc AB is 10.


 (A) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
 (B) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
CORRECT  (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
 (D) EACH statement ALONE is sufficient.
 (E) Statements (1) and (2) TOGETHER are NOT sufficient.
Information Given
O is the center of the circle.

Find the Real Information
Line AC is a [[chord]], and is the [[diameter]] of the circle.

Question Asked
What is the measure of angle ACB?

Find the Real Question
What is the radius OR circumference AND what is the arc length of BC?

Translate the Statements
(1) This statement gives us the radius; we also need to know the [[arc]] length, which can only be determined by knowing the measure of the [[central angle]].
Insufficient.
Eliminate A and D.

(2) This statement gives us the arc length of AB but not the radius or circumference.
Insufficient.
Eliminate B.

(1) + (2) Combined we are given the radius and an arc length. Given this information there are two ways to determine angle ACB.
- By definition the measure of an inscribed angle is one-half the measure of the corresponding arc length. Since we have the arc length and can determine the circumference of the circle (from the given radius), we can find the proportional measure of the angle using the arc-sector formula and then taking half that angle measure.

- If we take half of the circumference and subtract arc length AB from it we will get arc length BC. Drawing a radius from O to B will create an isosceles triangle. Again using the arc-sector formula we can determine the measure of the inscribed angle BOC. Since the triangle is isosceles, and the two remaining angles of the triangle are equal, we can then determine angle ACB. Sufficient.  Eliminate E.

Choose C.
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