Question

PQRS is a parallelogram. X and Y are mid-points of sides PQ and RS respectively. If W and Z are point on intersection of SX and PY and XR and YQ respectively, then show that ar(triangle YWZ) = ar(triangle XWZ)

Answer 1

PQRS is a parallelogram
$\Delta$ PYQ and PQRS are standing on same base PQ and between 2 parallels PQ and RS
$\Rightarrow ar\left(\Delta PYQ\right)=\frac{1}{2}ar\left(PQRS\right)$
Similarly $ar(OS\times R)=\frac{1}{2}ar\left(PQRS\right)$
$\Rightarrow ar\left(\Delta PYR\right)=ar\left(\Delta SXR\right)$
$ar\left(\Delta YWZ\right)+ar\left(WZQP\right)=ar\left(\Delta XWZ\right)+ar\left(SWZR\right)$
$\Rightarrow ar\left(\Delta YWZ\right)=ar\left(\Delta XWZ\right)$.