Question

Prove that the area of any quadrilateral with
perpendicular diagonals =1/2 × Product of
diagonals.​

Answer 1

Step-by-step explanation:

A quadrilateral ABCD in which  the diagonals AC and BD intersect at O assume that AO = OC , BO = OD . AC perpendicular BD.

To prove : ABCD is a rhombus.

Proof : Since the diagonals  AC and BD of quadrilateral ABCD bisect each other  at right  angles.

Therefore,  AC is the perpendicular bisector of the segment BD.

A and C both are equidistant from B and D.

AB = AD and CB = CD        ... (1)

Also , BD is the perpendicular bisector of line segment  AC.

AD = DC        ... (2)

From (1) and (2), 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.

now it is proved tha ABCD is a rombus and area of Rombus =1/2 product of diagonals.

Hence, proved