If AB is a diameter of the circle shown, what is the area of the circl

1. If BD = 3, diameter of the circle could be found as diameter is the hypotenuse of the triangle for which the other 2 sides are known.
With diameter, area of circle could be found.
So, 1 is sufficient to answer.
2. AC = sqrt(d^2-1) and BD = sqrt (d^2-4). Now AC/BD is given. So d could be found. So area can be found. 2 is sufficient to answer.

Hence the answer should be D.

Official Explanation

To find the area of the circle, we need to find the diameter, AB. This and only this will be sufficient. There is a key factoid for this question, which is that any triangle inscribed in a circle with the diameter as a side is a right triangle. It's quite a random thing to have to know for the test, but it is included in the math standards in the Official Guide. It means that we can get the diameter using the Pythagorean Theorem, if we have both other sides of either triangle.

Statement (1) gives us such a situation, so it's sufficient.

Statement (2) gives us one of the missing sides in terms of the other missing side. This seems insufficient at first blush, but let's consider the n equations, n variables rule. By the Pythagorean Theorem, we have two equations for the two triangles:

AC^2 + BC^2 = AB^2

AD^2 + BD^2 = AB^2

Substituting in the known quantities and Statement (2), we get

\((\frac{2\sqrt{3}}{3}*BD)^2 + 1^2 = AB^2\)

\(2^2 + BD^2 = AB^2\)

It turns out that we have two distinct equations and only two variables. We can substitute BD^2 and thus solve for AB^2 and we will be able to obtain the area. Statement (2) is therefore sufficient.

The correct answer is (D).

Video Explanation

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