swerve wrote:

In the figure above, \(ABCD\) is a rectangle. What is the area of the semi-circular region with center \(O\) and diameter \(BC\)?

1) \(\frac{BC}{AB}=\frac{3}{4}\)
2) \(BD=25\)

The OA is C

Source: Manhattan Prep

To get the area, we must know either BO or BC.

Let's take each statement one by one.

1) \(\frac{BC}{AB}=\frac{3}{4}\)

Ratio value can't help as it would not render a unique answer. Insufficient.

2) \(BD=25\)

Can't get BC with the help of BD. Insufficient.

(1) and (2) together

From (1), say BC = 3x and AB = CD = 4x

Since ABCD is a rectangle, ∆BCD is a right-angled triangle.

Thus, BC^2 + CD^2 = BD^2

(3x)^2 + (4x)^2 = (25)^2

=> x = 5 => CD = 15. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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