In the diagram above, what is the circumference of the circle?

1) The area of triangle is \(16\sqrt{3}\).

2) \(\angle B=60^{o}\).

The OA is C

**Source: Magoosh**

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**A**

**B**

**C**

**D**

**E**

We see that /_A = 90Âº; thus, âˆ†ABC is a right-angled triangle. And we know that only the diameter subtends an angle of 90Âº at the circumference of the circle. Thus, BC is the diameter. If we get the value of BC, we will get the value of circumference of the circle.

Let's take each statement one by one.

1) The area of triangle is \(16\sqrt{3}\).

=> 1/2*AB*AC = 16âˆš3 => AB*AC = 32âˆš3. We can't get the value of BC. Insufficient.

2) \(\angle B=60^{o}\).

=> âˆ†ABC is a 30-60-90 triangle. Thus, AB : AC : BC :: 1 : âˆš3 : 2. Since we do not have the actual value of any of the sides, we can't get the actual value of BC. Insufficient.

(1) and (2) together

From AB : AC : BC :: 1 : âˆš3 : 2, we have AC = âˆš3*AB and BC = 2*AB.

From AB*AC = 32âˆš3 and AC = âˆš3*AB, we have AB*AC = 32*âˆš3 = AB*(âˆš3*AB)

=> AB = âˆš32

=> BC = 2*âˆš32 = 8âˆš2.

Circumference of the circle = Ï€*BC = Ï€*8âˆš2. Sufficient.

The correct answer: C

Hope this helps!

-Jay

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