SSC CHSL 10 Jan 2017 Shift 3 Paper

In ΔAPQ and ΔABC
∠PAQ = ∠BAC (common)
∠APQ = ∠ABC (Alternate interior angles)
∠AQP = ∠ACB (Alternate interior angles)
=> ΔAPQ ∼ ΔABC
=> Ratio of Area of ΔAPQ : Area of ΔABC = Ratio of square of corresponding sides = ${\left(\mathit{A}\mathit{P}\right)}^2:{\left(\mathit{A}\mathit{B}\right)}^2$
⇒ $\frac{9}{16}=\frac{{\left(\mathit{A}\mathit{P}\right)}^2}{{\left(\mathit{A}\mathit{B}\right)}^2}$
⇒ $\frac{\mathit{A}\mathit{P}}{\mathit{A}\mathit{B}}=\sqrt{\frac{9}{16}}=\frac{3}{4}$
Also, PB = AB - AP
⇒ $\frac{\mathit{A}\mathit{P}}{\mathit{P}\mathit{B}}=\frac{3}{4-3}=\frac{3}{1}$