Question

If the diagonals AC and BD of a quadrilateral ABCD intersect at O such that AO. OD $=$ OB.OC, then the quadrilateral is a :

• A. parallelogram
• B. trapezium
• C. rectangle
• D. square

Answer 1

The correct option is A parallelogram

The diagram is in the attached image.

Draw a line EO such that

EO || AB

$\frac{DE}{EA}$ = $\frac{DO}{BO}$ ….(i) [ Proportionality Theorem]

Also, $\frac{AO}{BO}$ = $\frac{CO}{DO}$ Given

$\frac{DO}{BO}$ = $\frac{CO}{AO}$ ...(ii)

From equation (i) and (ii), we get

$\frac{DE}{EA}$ = $\frac{CO}{AO}$

By using converse of Basic Proportionality Theorem, EO || DC also EO || AB

AB || DC.

Hence, quadrilateral ABCD may be a trapezium with AB || CD or may be a parallelogram.

So correct option can be both A and B