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In the figure below, angle ABC is bisected by BF. If AD = 3 , BD = 5 , and BC = 10 , find

the fractional part of triangle ABC that triangle BFD occupies.

In the figure below, angle ABC is bisected by BF. If AD = 3 , BD = 5 , and BC = 10 , find-example-1

1 Answer

3 votes

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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In the figure below, angle ABC is bisected by BF. If AD = 3 , BD = 5 , and BC = 10 , find-example-1