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?
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.