Brent@GMATPrepNow wrote:

If ABCD is a square, and XYZ is an equilateral triangle, then the area of the square is how many times the area of the triangle?

A) (4√3)/3
B) (8√3)/3
C) 2√6
D) (16√3)/9
E) (16√2)/3

If ∆XYZ is EQUILATERAL, then each angle is 60°
So, if we draw a line from the center to a vertex, we'll get two 30° angles....


Now drop a line down like this to create a SPECIAL 30-60-90 right triangle


Since the base 30-60-90 right triangle has lengths 1, 2 and √3, let's give the triangle these same measurements...


IMPORTANT: This means the circle's radius = 2, which means the circle's DIAMETER = 4
Notice that the circle's diameter = the length of one side of the square
So, each side of the square has length 4, which means the area of the square = (4)(4) = 16

Okay, now let's determine the area of the triangle

Since we know that one side of the special 30-60-90 right triangle has length √3...

.... we know that the length of one side of the equilateral triangle = 2√3

This allows us to apply a special area formula for equilateral triangles:
Area of equilateral triangle = (√3)(side²)/4

So, the area of ∆XYZ = (√3)(2√3)²/4
= (√3)(12)/4
= 3√3

The area of the square is how many times the area of the triangle?
Answer = 16/3√3
Check the answer choices...not there!

Multiply top and bottom by √3 to get: (16√3)/9
Answer: D

Cheers,
Brent