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

Make a figure in the co-ordinate plane and assume that the smaller base of the trapezoid is 2b.

Draw the line y=h for -b<=x<=b

The slope of the diagonal will be 1 and the equation of the diagonal of the trapezoid will be y=x-(h-b) and will intersect the x axis at x=h-b.

That is, the larger base of the trapezoid is 2(h-b)

The area of the trapezoid is (h/2)[2b + 2(h-b)] = h²

Note: there is probably a nice geometric way to solve this problem, but I could not see it...therefore, I resorted to analytic geometry which gives the solution quickly.