Question

In the figure shown here, QS = SR, QU = SU, PW = WS and ST || RV. What is the value of \(\frac{\text{Area of ΔPSX}}{\text{Area of ΔPQR}}\) ?

(a) \(\frac{1}{5}\) 

(b) \(\frac{1}{3}\)

(c) \(\frac{1}{6}\)

(d) \(\frac{1}{7}\)

Answer 1

(a) \(\frac{1}{5}\)

Area (ΔPSX) = Area (PUS) – Area (SUX) 

In ΔPXS, WY || SX

⇒ \(\frac{PY}{YX}\) = \(\frac{PW}{WS}\) = 1            (Given, PW = WS)

⇒ PY = YX

In ΔRUY, SX || RY = \(\frac{UX}{XY}\) = \(\frac{US}{SR}\) = \(\frac{1}{2}\)

(∵ QS = SR and QV = US)

= UX = \(\frac{1}{2}\) (XY)

∴ In ΔPUS, UX = \(\frac{1}{2}\)XY = \(\frac{1}{2}\) \(\big(​​\frac{1}{2}PX\big)\) = \(\frac{1}{4}\) PX

PU = UX + PX = \(\frac{1}{4}\) PX + PX = \(\frac{5}{4}\)PX

∴ \(\frac{\text{Area of ΔSUX}}{\text{Area of ΔPUS}}\) = \(\frac{1}{5}\)

Now Area of ΔPUS = \(\frac{1}{4}\)(Area of ΔPQR)

Area of ΔSUX = \(\frac{1}{4}\) x \(\frac{1}{5}\) x Area of ΔPQR

= \(\frac{1}{20}\)Area of ΔPQR

∴ \(\frac{\text{Area of ΔPSX}}{\text{Area of ΔPQR}}\) = \(\frac{\frac{1}{4}Area\,of\,ΔPQR-\frac{1}{20}Area\,of\,ΔPQR}{Area\,of\,ΔPQR}\)

= \(\frac{5-1}{20}\) = \(\frac{1}{5}\).