In the circle above with center O, what is the length of

[BTGmoderatorLU]

Source: Veritas Prep



In the circle above with center O, what is the length of minor arc BC?

$$(1)\ \text{Diameter}\ AC = 24$$
$$(2)\ \text{Chord}\ AB = 12 \sqrt{3}$$
The OA is D.

[ceilidh.erickson]

Since we know that minor arc BC is a 60* arc (corresponding to a 30* interior angle), then we know that it makes up 1/6 of the circumference of the circle. If we can find the circumference, we'll know the value of arc BC.

(1) diameter AC = 24
Given the diameter, we'll be able to find the value of the circumference. Sufficient.

$$(2)\ \text{Chord}\ AB = 12 \sqrt{3}$$
Since AC is a diameter, we can create a 30-60-90 triangle by drawing a line from B to C. Since 30-60-90 triangles always have the same ratio of side lengths, knowing the side length of AB will allow us to infer that the diameter AC is 24. As with statement 1, knowing the diameter is sufficient to tell us the circumference, which is sufficient to tell us the length of arc BC.

The answer is D.