# GMAT OFFICIAL GUIDE DS – In the figure above, points A, B, C, D, and E lie…

## Solution:

Let’s sketch the diagram provided in the problem.

We are given that B is the center of the smaller circle and that C is the center of the larger circle. Thus, the radius of the smaller circle is line segment AB (or BD), and the radius of the larger circle is line segment AC (or CE). We need to determine the area inside the larger circle but outside the smaller circle. In order to do so, we would subtract the area of the smaller circle from the area of the larger circle.

The formula for the area of a circle is A = [Symbol]r^2. Thus, in order to determine the area inside the larger circle and outside the smaller circle, we need only to know the radii of the two circles.

Statement One Alone:

AB = 3 and BC = 2

From the information in statement one we can determine the radius of the larger circle and the radius of the smaller circle.

radius of smaller circle = AB = 3

radius of larger circle = AC = AB + BC = 3 + 2 = 5

Because we now have values for the radii of the two circles, we can determine the areas of both circles. It is not necessary to perform the calculation; however, since it’s not difficult to determine the area, the calculation is:

area = [Symbol]r^2

area of smaller circle = [Symbol](3)^2 = 9[Symbol]

area of larger circle = [Symbol](5)^2 = 25[Symbol]

Thus, the area inside the larger circle and outside the smaller circle is 25[Symbol] – 9[Symbol] = 16[Symbol].

Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

CD = 1 and DE = 4

From the information in statement two, we can see that the radius of the larger circle = CE = CD + DE = 5. From the given information we also know that AD is the diameter of the smaller circle. Because we know that the radius of the larger circle is 5, we know that the diameter of the larger circle, or line segment AE = 10. Thus, the diameter of the smaller circle is AD = AE – DE = 10 – 4 = 6 and, therefore, the radius of the smaller circle is 3. Since we have determined the radii of both circles, we have enough information to determine the areas of both circles.

Statement two alone is sufficient to answer the question.