Question

Question 7 (ii)
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:
$\left(ii\right)ar\left(RQC\right)=\frac{3}{8}ar\left(ABC\right)$

Answer 1

CR is a median of $\Delta CAP$,
$\Rightarrow ar\left(ARC\right)=\frac{1}{2}ar\left(CAP\right)=\frac{1}{2}\times \frac{1}{2}ar\left(ABC\right)=\frac{1}{4}ar\left(ABC\right)$ ------ (1)
Since RQ is median of $\Delta RBC$,
$ar\left(\Delta RQC\right)=\frac{1}{2}ar\left(\Delta RBC\right)$
$=\frac{1}{2}\left\{ar\right(\Delta ABC)-ar(\Delta ARC\left)\right\}$
$=\frac{1}{2}\left\{ar\right(\Delta ABC)-\frac{1}{4}ar(\Delta ABC\left)\right\}$
$=\frac{3}{8}ar\left(\Delta ABC\right)$
Hence,
$ar\left(\Delta RQC\right)=\frac{3}{8}ar\left(\Delta ABC\right)$