Question

The diagonals of a parallelogram $ABCD$ intersect at $O$. A line through O intersect AB at X and DC at Y. Identify the correct option

• A. $OX=OY$
• B. $OX>OY$
• C. $OX<OY$
• D. $OX-OY=0$

Answer 1

Answer: (A)

$OX=OY$

$\Rightarrow$ In given figure ABCD is a parallelogram.

$\therefore$ AB$\parallel$DC

$\Rightarrow$ Also AC is a transversal of AB$\parallel$DC.

$\therefore$ $\angle 1=\angle 2$ [Alternate interior angles]

$\Rightarrow$ Now in $△$AXO and $△$CYO, we have

$\Rightarrow$ $\angle 1=\angle 2$ [Alternate interior angles]

$\Rightarrow$ $\angle 3=\angle 4$ [Vertically opposite angles]

$\Rightarrow$ $CO=OA$ [Diagonals bisect each other]

$\therefore$ $△AXO\cong △CYO$ [ASA Criteria]

$\therefore$ $OX=OY$ [CPCT]