Question

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

• A. $\frac{1}{2}$
• B. $\frac{2}{3}$
• C. $\frac{3}{5}$
• D. $Noneoftheabove$

Answer 1

The correct option is A $\frac{1}{2}$

$\Delta APQ$ ~ $\Delta ACB,BC=2PQ$
and BC || PQ
$\Rightarrow$ AF = 2AE
$\Rightarrow$ AE = EF
Again $\Delta RGH$ ~ $\Delta RPQ$
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)
$then\frac{Areaof\Delta PQR}{Areaof\Delta ABC}=\frac{\frac{1}{2}\times PQ\times h}{\frac{1}{2}\times BC\times h}=\frac{PQ}{BC}=\frac{1}{2}$