geometry

The bases of an isosceles trapezoid have lengths that differ by 30. The two legs of the trapezoid have a length of 28, and the two diagonals of the trapezoid have a length of 52. What is the area of the trapezoid?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Height      h = sqrt(28² - 15²) 

Shorter Base      SB = sqrt(52² - h²) - 15

Longer Base      LB = SB + 30

Trapezoid area       A = 1/2 (SB + LB) * h

Let  the shorter  base =  B

Let the longer base =  B + 30

1/2  the  difference in the bases  =  (B + 30  - B)  /2  =  15

We  can  find the height of the trapezoid as    sqrt ( 28^2 - 15^2)  = sqrt (559)

And we can find the length of the shorter base   as 

sqrt  [ 52^2 - (15+ B)^2]  = sqrt (559)

2704  - B^2  - 30B - 225   =  559

B^2  - 30B  -  1920  =  0

B = sqrt (2145 )  -  15

And the longer base =  sqrt (2145) + 15

Area = (1/2) sqrt (559)  (  2sqrt (2145) )   =     sqrt (559) sqrt (2145)   ≈  1095 units^2