ABC is an equilateral triangle. Let PQRS be a square inscribed in it such that P is on AB and Q is on AC. Which of the following is/are correct ? I. AP : PB = 4 : 3 II. \sqrt{3}AB = (2 + \sqrt{3})PQ Select the answer using the code given below :

Explanation

Let square side be x. Triangle APQ is equilateral, so AP = x. Triangle PBS is right-angled at S with \angle B = 60^\circ, so PB = \frac{2x}{\sqrt{3}}. Thus AB = AP + PB = x + \frac{2x}{\sqrt{3}} = \frac{(2+\sqrt{3})x}{\sqrt{3}}. This gives \sqrt{3}AB = (2+\sqrt{3})x, making II correct. The ratio AP:PB is \sqrt{3}:2, so I is incorrect.

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