We can plug in the answers, which represent AE:AD.
Anytime √3 appears in the problem or in the answer choices,
LOOK FOR A 30-60-90 triangle.
The most likely answer choice is
A, which says that AE:AD = 1:√3, implying that ∆ADE is a 30-60-90 triangle whose sides are proportioned 1:√3:2.
When a rectangle is inscribed in a circle, DRAW THE DIAGONAL, since the diagonal of the rectangle = the diameter of the circle.
Thus, in the figure above, BD is both the diagonal of rectangle ABCD and the diameter of the circle.
If ∆ADE is a 30-60-90 triangle, then so is ∆BCD, since it is given that angle ADE = angle BDC.
Thus, the sides of ∆BCD are proportioned 1:√3:2.
Since the shortest side is √3, and 1:√3:2 = √3 : 3 : 2√3, we get:
BC = √3, CD = 3, and BD = 2√3.
Area of the circle:
Since BD = 2√3, r=√3.
Area = πr² = π(√3)² = 3π.
Area of rectangle ABCD:
CD*AD = 3√3.
Thus:
Circle:triangle = 3π : 3√3 = π:√3.
Success!
The correct answer is
A.
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