GMATPrepNow wrote:
In the above figure, the small circle and big circle have diameters of 3 and 6 respectively. If AB||CD, and both circles share the same center, what is the area of the shaded region?
A) 9/2 + 2π
B) 9/2 + 3π
C) 3√3 + 3π
D) 6√3 + 2π
E) (9/2)√3 + 3π
*kudos for all correct
solutionsIf we recognize that the hypotenuse of the blue right triangle below is twice the length of one side, we can see that this is a 30-60-90 SPECIAL TRIANGLE
In the base triangle (on the right), the side opposite the 30° has length 1.
So, the magnification factor of the triangle in the question is 3/2 (i.e., the triangle in the diagram is 3/2 times the size of the base triangle)
So, the length of the 3rd side = (3/2)(√3) = (3√3)/2
Let's add this to our diagram...
From here, we can calculate the area of the shaded triangle.
The length of the base = (3√3)/2 + (3√3)/2 = 3√3
Area = (base)(height)/2
= (3√3)(3/2)/2
=
(9√3)/4Let's add this to our diagram...
Important: we earlier learned that the triangle in the first image is a 30-60-90 special triangle.
In fact, there are four such special triangles in the diagram.
So, let's add the 30° angles
Let's now find the area EACH sector.
Area of sector = (central angle/360°)(π)(radius²)
= (60°/360°)(π)(3²)
= (1/6)(π)(9)
=
3π/2So, the area of the shaded region =
(9√3)/4 +
(9√3)/4 +
3π/2 +
3π/2= (9√3)/2 + 3π
Answer: E
Cheers,
Brent
_________________
If you enjoy my solutions, you'll love my GMAT prep course.