Can anyone solve this level 5, 10% accuracy geometry problem?

Question

https://gmatclub.com/forum/download/file.php?id=88864 GMAT-Club-Forum-bc6vvr0c.png

pintukr · Answer

Have geometry questions started figuring in the GMAT syllabus again ?

Sajjad1994 · Answer

Hi

This is a GRE (quantitative comparison) question.

Sajjad1994 · Answer

Explanation

Interpret Quantity A

The area of the circle is  πr^2

The area of the inscribed regular hexagon is given as x.

Therefore, Quantity  A=πr^2−x, , which is exactly the area  inside the circle but outside the hexagon .

Break that region into congruent pieces. A regular hexagon divides the circle into  6 equal circular segments , one along each side of the hexagon.

Let the area of one such segment be T, then

πr^2−x=6T.

Relate the shaded region S to one segment

The second polygon is a regular 12-gon.

Each side of the 12-gon subtends

\frac{360}{12}=30∘ ,

while each side of the hexagon subtends

\frac{360}{6}=60∘ ,

Since the polygons share six vertices,  each side of the hexagon is replaced by two sides of the dodecagon .

Thus, one hexagon segment is split into  two identical smaller curved regions , one of which is the shaded region S.

Hence,

T=2S

So

πr^2−x=6T=6(2S)=12S

Therefore,

Quantity A  =12S 
  Quantity B  =24S

Since
 24S>12S

Answer: B

Prep Toolkit