Question

If a quadrilateral is constructed with equal diagonals bisecting each other, then it is:

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

Answer 1

Answer: (A), (C)

The correct options are
A square
C rectangle

A quadrilateral $ABCD$ is constructed by bisecting the equal diagonals $AC$ and $BD$ each other.

In $△AOB$ and $△COD$,

$AO=OB=DO=OC$

$\angle AOB=\angle DOC$ ....... (Vertically opposite angles)

$⟹△AOB\sim △COD$ ......... $\left(1\right)$

$\therefore \angle DCO=\angle OAB$ and $\angle DCA=\angle CAB$

These are the angles formed by transversal $AC$ intersecting $AB$ and $CD$.

$\therefore$ $AB\parallel CD$ and $AD\parallel BC$.

Thus, the quadrilateral formed is rectangle.

Similarly, from $\left(1\right)$

$△AOD\sim △BOC$

$\because$ the diagonals are equal

$\therefore$ All four triangles are similar and have same area.

Thus, the quadrilateral formed is a square.

Hence, either a rectangle or a square.