Question

What is the area of an isosceles trapezoid if the length of an altitude is h, and diagonals are perpendicular to each other?

Answer 1

In order to answer that question, you have to work with right triangles in this ABCD trapezoid. Because of diagonals are perpendicular, we say that ΔAOD and  ΔBOC are right-isosceles triangles. ∠OBC= ∠FBD = 45°. Then, BF = FD = h. Similarly, in triangle ACE ∠ACE = 45°. From here, CE = AE = h. 

The area of this trapezoid is $A= \frac{AD+BC}{2}h$. We need to find AD and BC in order to calculate this area. Since we know that BC = FE. AE + FD = AD + BC. We know that AE + FD = 2h. 

The area is $A= \frac{2h}{2} h = h^{2}$