In the diagram above, what is the circumference of the

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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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by [email protected] » Tue Jun 11, 2019 9:38 pm

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swerve wrote:Image

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
We see that /_A = 90º; thus, ∆ABC is a right-angled triangle. And we know that only the diameter subtends an angle of 90º 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 = 16√3 => AB*AC = 32√3. We can't get the value of BC. Insufficient.

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

=> ∆ABC is a 30-60-90 triangle. Thus, AB : AC : BC :: 1 : √3 : 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 : √3 : 2, we have AC = √3*AB and BC = 2*AB.

From AB*AC = 32√3 and AC = √3*AB, we have AB*AC = 32*√3 = AB*(√3*AB)

=> AB = √32

=> BC = 2*√32 = 8√2.

Circumference of the circle = π*BC = π*8√2. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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