Question

$ABCD$ is a quadrilateral whose diagonal $AC$ divides it into two parts, equal in area, then $ABCD$

• A. is a rectangle
• B. is always a rhombus
• C. is a parallelogram
• D. can be any quadrilateral

Answer 1

The correct option is D can be any quadrilateral

Given:

$ABCD$ is a quadrilateral with its diagonal $AC$ dividing it into two parts, equal in area.

To find:

Type of quadrilateral of $ABCD$

Construction:

The perpendiculars $p$ & $q$ are dropped from $B$ & $D$ respectively on $AC$

Solution:

$ar\Delta ACD=\frac{1}{2}AC\times q$ & $ar\Delta ABC=\frac{1}{2}AC\times p$

Given that $ar\Delta ACD=ar\Delta ABC$

$\Rightarrow \frac{1}{2}AC\times q=\frac{1}{2}AC\times p$

$\Rightarrow p=q$

So, this is the only and only criterion here.

So $ABCD$ need not be any of the options $A,B$ or $C$

Hence, option D.