Question 1148964: ABCD is a trapezoid in which AD:BC = 2:5. If the area of triangle AMD is 624cm2, find the area of the trapezoid in cm2.
Diagram: https://imgur.com/a/kqz5EdC
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Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Triangles AMD and CMB are similar. Since the bases are in the ratio 2:5, their heights are in the ratio 2:5, and their areas are in the ratio 2^2:5^2 = 4:25.
So, given that the area of AMD is 624, we know the area of CMB is 624*(25/4) = 3900.
Triangles AMD and ABD share base AD; and the ratio of their altitudes is 2:(2+5) = 2:7. So the area of triangle ABD is 624*(7/2) = 2184; and then the area of triangle ABM is 2184-624 = 1560.
By a similar argument, the area of triangle DMC is also 1560.
So the area of the trapezoid is
624+3900+2(1560) = 7644 cm^2.
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Added in response to the reader's question....
The ratio of similarity between triangles AMD and CMB is 2:5, not 2:7.
2:7 is the ratio of the height of triangle AMD to the height of the trapezoid.
2:7 is also the ratio between the heights of triangles AMD and ABD; and between the heights of triangles AMD and DMC.
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