Question

ABCD is a parallelogram in which X and Y are the mid-points of AB and CD. AY and DX are joined which intersect each other at P. BY and CX are also joined which intersect each other at Q. Show that PXQY is a parallelogram

Answer 1

$\sf \bigstar \:Question :-$

ABCD is a parallelogram in which X and Y are the mid-points of AB and CD. AY and DX are joined which intersect each other at P. BY and CX are also joined which intersect each other at Q. Show that PXQY is a parallelogram

To prove :-

PXQY is a parallelogram

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solution:-

AB//DC

=> AX //YC

$\sf \: \frac{1}{2} ab = \frac{1}{2} dc$

AX = YC

therefore , AX//YC & AX=YC

AXCY is a parallelogram

AY// XC or PY //XQ

AB=DC

$\sf \: \frac{1}{2} ab = \frac{1}{2} dc = > xb = dy \\ \sf \therefore \: xb \parallel \: dy \: and \: xb = dy$

XBYD is a //gm

XD//BY or XB//QX

thus , PY//XQ and XP//QY

hence , its proved that PXQY is a parallelogram