Question

P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP,
Show that $ar\left(\Delta PBQ\right)=ar\left(\Delta ARC\right)$.

Answer 1

Construction: Join PQ, RQ and RC

Proof:
(3)
CR is a median of
$\Delta CAP$
$\Rightarrow ar\left(\Delta ARC\right)=\frac{1}{2}ar\left(\Delta CAP\right)$ ---- (1)
Also, CP is a median of
$\Delta ABC$
$\Rightarrow ar\left(\Delta CAP\right)=ar\left(\Delta CBP\right)$ ------ (2)
From (1) and (2), we get,
$\Rightarrow ar\left(\Delta ARC\right)=\frac{1}{2}ar\left(\Delta CBP\right)$ ----- (3)
PQ is a median of
$\Delta PBC$,
$\Rightarrow ar\left(\Delta CPB\right)=2ar\left(\Delta PBQ\right)$ ----- (4)
From (3) and (4), we get,
$\Rightarrow ar\left(\Delta ARC\right)=ar\left(\Delta PBQ\right)$ ----- (5)
Hence,
$ar\left(\Delta ARC\right)=ar\left(\Delta PBQ\right)$