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
