Answer:
5/24
Explanation:
In the figure shown with angle bisector BF and AD=3, BD=5, BC=10, you want the ratio of the area of triangle BFD to the area of triangle ABC.
Angle bisector
The angle bisector BF divides the sides of triangle BCD proportionally. This means ...
DF/CF = DB/CB
DF/CF = 5/10 = 1/2 . . . use given numbers
CF = 2DF . . . . . . . . . . . multiply by 2·CF
The length DC is then ...
DC = DF +CF = DF +2DF = 3DF
and
DF = 1/3·DC
Area
The area of a triangle is proportional to the base and the height.
In this figure, the base of ∆BFD is BD = 5, and the base of ∆ABC is AB = (3+5) = 8. That is, the base of ∆BFD is 5/8 of the base of ∆ABC.
The height of ∆BFD is proportional to the length DF, and the height of ∆ABC is proportional to the length DC. This means the height of ∆BFD is 1/3 the height of ∆ABC.
Then the area ratio is ...
area ∆ABC is proportional to (base ABC)(height ABC)
area ∆BFD is proportional to (base BFD)(height BFD)
= (5/8·base ABD)(1/3·height ABC) = 5/24·(base ABC)(height ABC)
The area of ∆BFD is 5/24 of the area of ∆ABC.
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Additional comment
The attached figure shows one instance of such a geometry. The numbers in color are the areas of the corresponding triangles. You can see they have the ratio 5/24.
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