Question

In the given figure, (not drawn to scale), P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC and QD is parallel to CP. In ΔARC, ∠ARC = 90º and in ΔPQS, ∠PSQ = 90º. The length of QS = 6 cm. What is the ratio of AP : PD? 

(a) 10 : 3 

(b) 2 : 1 

(c) 7 : 3 

(d) 8 : 3

Answer 1

(c) 7 : 3.

PQ || AC

⇒ \(\frac{CQ}{QB}=\frac{AP}{PB}=\frac{4}{3}\)                  .......(i)

 Also, QD || CP

⇒ \(\frac{PD}{DB}=\frac{CQ}{QB}=\frac{4}{3}\)                   (From (i))

\(\frac{PD}{DB}=\frac{4}{3}\) ⇒ \(\frac{PD}{DB}\) = \(\frac{PD}{PD+DB}\) = \(\frac{4}{4+3}\) = \(\frac{4}{7}\)            ...........(ii)

∴ From (i) and (ii)

\(\frac{AP}{PB}=\frac{4}{3}\) and \(\frac{PB}{PD}=\frac{7}{4}\)

∴ \(\frac{AP}{PB}\times\frac{PB}{PD}\) = \(\frac{4}{3}\) x \(\frac{7}{4}\) ⇒ \(\frac{AP}{PD}=\frac{7}{3}\) ⇒ AP : PD = 7 : 3.