Did you try out our GMAT Data Sufficiency practice question? If not, take a couple minutes now to give it a try before reviewing the explanation.
Now, let’s get down to brass tacks on the Geometry and DS skills you need to solve this one…
One way to find the area of a quadrilateral is to divide it into triangles and add the areas of the triangles, which can be found using the formula for the area of a triangle: (1/2)(Base)(Height).
If you add dashed lines to the diagram connecting points A and C and points B and D, you will see that the quadrilateral is composed of 4 right triangles:
You can see that one side of each triangle is a radius of one of the circles; for example, AB is a radius of circle A and is the hypotenuse of triangle ABE. Also, you’ll notice that triangles ABE and ADE are identical, as are triangles BCE and CDE. In order to be sufficient, the statements will have to provide you with the bases and heights of these triangles. The bases are AE and EC, and the heights are BE and DE (which are identical).
Statement (1) gives you the radius of each circle, 10 and 5. So, you know the hypotenuse of each triangle but not the base or height. In fact, there is no way to find the bases and heights. For example, consider triangle ABE. The area is (1/2)(B)(H), or (1/2)(AE)(BE). The perimeter is:
AB + AE + BE = 10 + AE + BE.
You have four unknowns (the area, the perimeter, AE and BE) and only two equations. So, you can’t find the information you need. Statement (1) is insufficient.
Statement (2) provides even less information, unfortunately. Now you don’t even know the radii of the circles, just the ratios of their areas. You can’t find the bases and heights of the triangles, so Statement (2) is also insufficient.
The statements are insufficient even when combined. Given the radii in Statement (1), the area of circle A is
and the area of circle C is
Based on this, you already know that the area of A is 300% greater than the area of C, meaning that Statement (2) is not providing any new information. Since Statement (1) by itself is insufficient and Statement (2) adds nothing new, the statements cannot be sufficient when they are combined.
Answer choice (E) is the correct choice.
Stay tuned – tomorrow I’ll share my advice on how to get the most out of this (and every) practice problem and answer explanation. Until then, happy studying!
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