trapezoid

Call the height of the trapezoid "x"   --->   AD  =  x

Since CD = 4, triangle(CDA) is a right triangle with CA  =  sqrt(4² + x²)  =  sqrt(16 + x²)

Since AB = 9, triangle(BAD) is a right triangle with BD  = sqrt(9² + x²)  =  sqrt(81 + x²)

 

The area of the trapezoid will be the area of triangle(CDA) + area triangle(CAB).

Let CA be the base of each of these triangles.

Since the diagonals are perpendicular, the sum of the heights of these triangles is BD.

 

Area of trapezoid(ABCD)  =  Area of triangle(CDA) + area triangle(CAB)  =  ½·sqrt(16 + x²)·sqrt(81 + x²).

 

Also, Area of trapezoid(ABC)  =  ½·x·(4 + 9)  =  (13/2)·x

 

Setting these two area equal to each other:  (13/2)·x  =  ½·sqrt(16 + x²)·sqrt(81 + x²).

--->      13x  =  sqrt(16 + x²)·sqrt(81 + x²).

--->   169x²  =  (16 + x²)·(81 + x²).

--->   169x²  =  1296 + 97x² + x⁴

--->           0  =  x⁴ - 72x² + 1296

--->           0  =  (x² - 36)²

--->           0  =  x² - 26

--->           x  =  6

 

Area  =  (13/2)·x  =  (13/2)·6  =  39