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Let ABC be a triangle, and let its angle bisectors be AD, BE, and CF which intersect at I. If DI=3, BD=4 and BI=6 then compute the area of triangle BID.

 Apr 28, 2024
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To find the area of triangle BID, we can use the formula for the area of a triangle given the length of one side and the lengths of the two adjacent sides to an angle. The formula is:

Area=12×side×adjacent side×sin(angle)

Given:
- DI=3
- BD=4
- BI=6

We need to find the angle at vertex B.

We know that the angle bisectors of a triangle divide the opposite side into segments that are proportional to the adjacent sides. Therefore, from the given information, we can set up the following proportions:

DIBD=DIDI+IB=34

34=33+IB

3+IB=4

IB=1

Now, we can find sin(B) using the Law of Sines in triangle BDI:

sin(B)=DIBI=36=12

Now, we can use the formula for the area of triangle BID:

Area=12×BD×DI×sin(B)

Area=12×4×3×12

Area=6square units

So, the area of triangle BID is 6square units.

 Apr 28, 2024

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