Question

$ABCD$ is a parallelogram $AP$ and $CQ$ are perpendicular drawn from vertices $A$ and $C$ on diagonal $BD$ (see figure) show that
$AP=CQ$

Answer 1

Given ABCD is a parallelogram AP and CQ are perpendicular drawn from verities A and C on BD

Then area of triangle $BDC$ =$\frac{1}{2}\times BD\times CQ$

And area of triangle $ABD$ =$\frac{1}{2}\times BD\times AP$

We know that diagonal BD divided ABCD is a parallelogram in two equal area

$\frac{1}{2}\times BD\times CQ=\frac{1}{2}\times BD\times AP$

$\Rightarrow CQ=AP$