# Suppose p,q,r,s are fixed real numbers such that a quadrilateral can be formed with sides p,q,r,s taken in a clockwise order. The vertices of the quadrilateral of maximum area lie on a circle. (True/False)

Jitender Singh IIT Delhi
8 years ago
Ans: True
Let ABCD be a quadrilateral with given sides p, q, r & s in clockwise order.
Area:
Area of triangle ABD + Area of triangle BDC
$\frac{1}{2}pq.sinA + \frac{1}{2}rs.sinC$
This value will be maximum when the angle A & C are commplementary.
$\angle A + \angle C = 180$
So the quadrilateral must be cyclic to have max. area.
Thanks & Regards
Jitender Singh
IIT Delhi