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

Dear Student,

Consider the figure as given below:


Given: PQRS is a parallelogram. X and Y are midpoints of PQ and RS respectively.
To Prove: $Ar∆\left(YWZ\right)=Ar∆\left(XWZ\right)$
Construction: Join WZ.
Proof: PQRS is a ​parallelogram.
$$\begin{gathered} \Rightarrow PQ\parallel RS \\ \Rightarrow PX\parallel RY \end{gathered}$$
X and Y are midpoints of PQ and RS respectively and PQ = RS.
$\Rightarrow PX=RY$
Since, one opposite side is parallel and equal so, PXRY is a ​parallelogram.
$\Rightarrow YW\parallel ZX ....\left(i\right)$
Similarly, XQYS is a ​parallelogram.

$$\begin{gathered} \Rightarrow YZ\parallel WX ....\left(ii\right) \\ From \left(i\right) and \left(ii\right) \end{gathered}$$
YWXZ is a ​parallelogram.
As, WZ is a diagonal and a diagonal divides the paralellogram in two congruent triangles.
$$\begin{gathered} \Rightarrow ∆YWZ\cong ∆XWZ \\ \Rightarrow Ar\left(∆YWZ\right)=Ar\left(∆XWZ\right) \end{gathered}$$
Hence, proved.
Regards,