M7MBA wrote: ↑

Tue May 12, 2020 6:49 am

Math.jpg

If the area of Rectangle \(ABCD\) is \(4\sqrt3,\) then what is the area of the square \(DEFG ?\)

A) \(\sqrt3\)
B) \(2\sqrt3\)
C) \(4\)
D) \(4\sqrt3\)
E) \(12\)

[spoiler]OA=E[/spoiler]

Solution:

We see that the area of rectangle ABCD is the product of the two legs of a 30-60-90 triangle (triangle ABC or triangle ADC). If the shorter leg of one of these two triangles is x, then its longer leg is x√3. So the area of rectangle ABCD is x * x√3 = x^2 * √3. Since we are given that the area of rectangle ABCD is 4√3, we see that x must be 2 (notice 2^2 = 4).

However, the side length of square DEFG is a leg of the triangle CDE (a 45-45-90 triangle), which in turn is the longer leg of triangle ADC; therefore, the side length of square DEFG is 2√3, and the area of square DEFG is (2√3)^2 = 4(3) = 12.

Answer: E