Question

If the diagonals of a quadrilateral bisect each other at right angles, then which type of quadrilateral is formed?

Answer 1

Since the diagonals AC and BD of quadrilateral ABCD bisect each other at right angles, AC is the perpendicular bisector of line segment BD.

$\Rightarrow$ A and C both are equidistant from B and D

$\Rightarrow$ AB = AD and CB = CD …(i)

($\because$ Locus of a point equidistant from two fixed points is the perpendicular bisector of the line segment joining those two points. Also, every point on the perpendicular bisector of the line joining two fixed points is equidistant from those two points).

Also, BD is the perpendicular bisector of line segment AC, since the diagonals AC and BD of quadrilateral ABCD bisect each other at right angles.

$\Rightarrow$ B and D both are equidistant from A and C.

($\because$ Locus of a point equidistant from two fixed points is the perpendicular bisector of the line segment joining those two points. Also, every point on the perpendicular bisector of the line joining two fixed points is equidistant from those two points).

$\Rightarrow$ AB = BC and AD = DC …(ii)

From (i) and (ii), we get

AB = BC = CD = AD

Thus, ABCD is a quadrilateral whose diagonals bisect each other at right angles and all four sides are equal.

Hence, ABCD is a rhombus.