How to find the area of the trapezium

Question

a trapezium

How can one find the area of trapezium with the help of the information given. Non trigonometric derivation expected

Answer 1

Let $DK$ be an altitude of $\Delta ADB$.

Thus, since $$DB=\sqrt{25^2-15^2}=20,$$ we obtain by calculating twice of the area of $\Delta ADB$: $$DK\cdot 25=15\cdot20,$$ which gives $DK=12.$

Now, $$AK=\sqrt{15^2-12^2}=9.$$ Thus, $$DC=25-2\cdot9=7$$ and $$S_{ABCD}=\frac{(25+7)12}{2}=192.$$