Question

PQRS is a square with mid points X and Y of sides SR and PQ respectively .Diagonal SQ intersects XY at O. Show that ar triangle SOX = triangle ar YOQ

Answer 1

PQRS is a square. So PQ = RS
Since X and Y are the mid-points of RS and PQ respectively,
The diagonal of the square SQ and XY bisect each other at O.
So XO = YO

In ΔSOX & ΔYOQ
$\angle SOX= \angle QOY$ (opposite angles)
XO = YO
$\angle OXS= \angle OYQ$ (both are right angles)

So ΔSOX $\cong$ ΔYOQ
Congruent triangles have equal area. 
So ar(ΔSOX) = ar(ΔYOQ) (proved)

Answer 2

Refer to attachment

Hope it helps