We have a trapezoid with bases $4$ and $9$, and diagonals $5$ and $12$. Determine the area of it and also the angle between the diagonals. The area is $15\sqrt{2}$; with cosine formula it is $90$ degrees, but I don't want to use it.
I have done every strategy I knew!
It will be great if you check it and help me.
As you saw, the Cosine Law tells us the angle. Let us see whether we can avoid the Cosine Law and use a more geometric approach.
Imagine the trapezoid in standard position, bottom side $9$, top side $4$. Now look at the top triangle and bottom triangle. They are similar, so the sides of the top triangle are $4\cdot \frac{5}{13}$ and $4\cdot \frac{12}{13}$. It follows (converse of Pythagorean Theorem) that the angle between the diagonals is $90$ degrees.
