In the above diagram, ∠QPT = ∠RST = 30°. If the area of ∆PTS = √12,

Since we're told PQRS is a SQUARE, we know that all 4 angles are 90 degrees.
So, if ∠QPT = ∠RST = 30°, then the two other angles are each 60°
If two of the angles in the triangle 60° each, then the third angle must also be 60° [since angles in a triangle add to 180°]
So, we now know that triangle PTS is an equilateral triangle
We get:


Area of equilateral triangle \(= \frac{\sqrt{3}}{4}(side)^2\)

Since we're told the area of ∆PTS = √12, we can write: \(\frac{\sqrt{3}}{4}(x^2)=\sqrt{12}\)

Multiply both sides by \(4\) to get: \(\sqrt{3}(x^2)=4\sqrt{12}\)

Divide both sides by \(\sqrt{3}\) to get: \(x^2=\frac{4\sqrt{12}}{\sqrt{3}}\)

Simplify numerator: \(x^2=\frac{8\sqrt{3}}{\sqrt{3}}\)

Simplify : \(x^2=8\)

Since the area of square PQRS \(= x^2\), we know that the area of square PQRS is \(8\)

Answer: C

Cheers,
Brent