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.
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.