Question

D is any point on AC in $ABC.$ Now, $P,Q,X,Y$ are the mid points of AB, BC, AD and DC respectively. Show that PX = QY.

Answer 1

In $△ABC,P$ and $Q$ are mid points of $AB$ and $BC$ respectively.

So, $PQ||AC$ and $PQ=\frac{1}{2}AC$ ..................(1) (Mid point theorem)

Similarly, $X$ and $Y$ are mid points of sides $CD$ and $AD$ respectively.

we get

$PQ||XY$ and $PQ=XY$

Hence, $PQXY$ is a parallelogram .... (pair of opposite sides is parallel and equal)

Now, $XY||AC$ and $QY||BD$

And diagonals in $PQXY$ are equal

Then $PX=QY$