This is a problem from the Regional Mathematics Olympiad, RMO 2015 Mumbai Region based on Diagonal of a Quadrilateral. Try to solve it.
Problem: Diagonal of a Quadrilateral
Let ABCD be a convex quadrilateral with AB = a, BC = b, CD = c and DA = d. Suppose and the area of ABCD is 60 square units. If the length of one of the diagonals is 30 unit, determine the length of the other diagonal.
(Solution suggested in class by Megha Chakraborty)
It is given that
Multiplying 2 to both sides we have
But sum of squares can be 0 if and only if each square is individually 0. This implies a = b = c = d. Hence the quadrilateral is a rhombus (in a special case, a square).
The area of a rhombus is where and are the diagonals. It is given that one of the diagonals is 30 and area is 60. Hence we have
Hence the length of the other diagonal is 4.