+128475 A B
16
P
25
D C
The area = 81
To see why
Triangles PAB and PDC are similar because angle PDC = angle PBA and angle PCD =angle PAB
Since the area of APB =16 and the area of DPC = 25....the scale factor is sqrt [ 16/25] =4/5 = 4 : 5
So AB = (4/5)CD and the height of APB = (4)/(4 + 5)*h = (4/9) h (where h is the height of the trapezoid)
Let the base of DPC = b = DC.....so the base of APB = (4/5)b = AB
Using triangle PAB
Area = (1/2) (AB) ( 4/9) h
16 = (1/2)(4/5)b * (4/9) h
32 = [ 16 / 45 ] bh
32 ( 45 /16) = bh
90 = bh
b = 90/h = DC
Then the base of PAB = (4/5) (DC) = (4/5) (90/h) = 72/h
So the area of the trapezoid is
(1/2) (h) (AB + DC) =
(1/2) (h) ( 72/h + 90/h) =
(1/2) ( h) (72 + 90) / h =
(1/2) (162) =
81
