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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
 #1
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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:
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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:

Now, we can find using the Law of Sines in triangle BDI:

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

So, the area of triangle BID is .

 Apr 28, 2024

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