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bar(EG)|| bar(DH), and the length of seg...

`bar(EG)|| bar(DH)`, and the length of segments `bar(DE)` and `bar(EF)` are as marked. If the area of `Delta EFG` is a, what is the area of `Delta DFH` in terms of a ?

A

`(4a)/(5)`

B

`(16a)/(25)`

C

`(16a)/(20)`

D

`(25a)/(16)`

Text Solution

Verified by Experts

The correct Answer is:
D
The only information that's in the question and not is the figure is that `bar(EG)` and `bar(DH)` are parallel and that the area of `Delta EFG` is a. That `bar(EG)` and `bar(DH)` are parallel tells you that `Delta EFG` and `Delta DEFH` are similar - they have the same angles, and sides are in proportion. Because the ratio of `bar(DF)` to `bar(EF)` is `(5)/(4)`, the ratio of any pair of corresponding sides will also be `(5)/(4)`. But that's not the ratio of the areas. Remember that the ratio of the areas of similar triangles is the square of the ratio of the sides. Here the side ratio is `(5)/(4)`, so the area ratio is `(5^(2))/(4^(2))=(25)/(16)`. If the area of the small triangle is a, then the area of the large one is `(25a)/(16)`.
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