Question

In the adjoining figure, P and Q are the mid-points of AC and AB. Also, PG = GR and HQ = HR. What is the ratio of the area of $\Delta PQR$: area of $\Delta ABC$?

• A. $\frac{1}{2}$
• B. $\frac{2}{3}$
• C. $\frac{3}{5}$
• D. None of these

Answer 1

The correct option is A

$\frac{1}{2}$

$\Delta$ APQ $\sim \Delta$ ACB, BC = 2PQ
and BC ||PQ
$\Rightarrow$ AF = 2AE
$\Rightarrow$ AE= EF
Again $\Delta$ RGH $\sim \Delta$ RPQ
and PQ=2GH
(By mid-point theorem)
$\Rightarrow$ RJ=2RK
$\Rightarrow$ RK=JK
But since EF = JK
$\therefore$ AE=EF=JK=RK
$\therefore$ RJ=RK+JK and AF=AE+EF
and RJ=AF=h(say),
then $\frac{Areaof\Delta PQR}{Areaof\Delta ABC}=\frac{\frac{1}{2}\times PQ\times h}{\frac{1}{2}\times BC\times h}=\frac{PQ}{BC}=\frac{1}{2}$