Since two opposite vertices of the above quadrilateral are right angles, the quadrilateral is cyclic and we are able to make use of Ptolemy’s theorem, which states that the product of the diagonals is equal to the sum of products of opposite sides.
We know the two diagonals, and two of the sides. Let’s call the other two sides a and b. The area we seek is ab/2.
By Pythagoras: a^2+b^2 = (10*sqrt(2))^2 = 200
By Ptolemy: 10a+10b=140*sqrt(2)
a+b=14*sqrt(2)
Let’s square both sides:
(a+b)^2=392
a^2+b^2+2ab=392
But we already know the value of a^2+b^2
200+2ab=392
ab=96
Area = ab/2 = 48 square units.
