Bunuel wrote:
In the diagram above, A is the center of the circle, angle BAD = 90°, and the area of triangle ABD is \(24\sqrt{3}\). What is the area of the circle?
(1) AB = 12
(2) AD = CD
Kudos for a correct solution.Attachment:
gdrtq_img1.png
VERITAS PREP OFFICIAL SOLUTION:In order to find the area of the circle, we would need the radius, AD.
Statement #1: AB = 12. Notice that AD is the base of triangle ABD. AB is the height, now known, and we know the area. If we know the height and the area, we can find the base by A = 0.5*bh. Therefore, we can find AD, which would allow us to find the area of the circle. This statement, alone and by itself, is sufficient.
Statement #2: We know that AD = AC, because all radii of a circle are equal. If CD also equals these two, then ACD is an equilateral triangle. That means that the angle ADC = 60°, which means that triangle ABD is a 30-60-90 triangle. If AD = x is the base, then this times the square-root of 3 is the height, and we could create an equation because we know the area:
\(24\sqrt{3}=\frac{1}{2}*x*x\sqrt{3}\)
This is an equation we could solve for x, which would allow us to find the area of the circle. This statement, alone and by itself, is sufficient.
Answer = (D)
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