Question

ABC is a trapizoid with AB parallel to CD . The diagonal AC and BD meet at P.if the area of ABP is 16 and the area of CDP is 25 , what is the area of the trapezoid?

Answer 1

It is easy to show that triangle APB and CPD are similar
The areas of similar triangles are proportional to the squares of their
corresponding sides,

then AB^2 : CD^2 = 16 : 25
AB:CD = 4:5
also the ratio of their heights should be 4:5

Let AB = 4x , then CD = 5x
Let the height of the smaller be 4y, and the height of the larger be 5y

area of the whole thing = (sum of parallel sides)(height)/2
= (4x + 5x)(4y+5y)/2 = (9x)(9y)/2 = 81/2 xy

we also know that (1/2)(4x)(4y) = 25
xy = 25/8

area of trapezoid = (81/2)(25/8) = 2025/16 = 126.5625

check my arithmetic, the answer seems a bit high.