In `DeltaABC, P,Q, R` are the feet of angle bisectors from the vertices to their opposite sides as shown in the figure. `DeltaPQR` is constructed

If `AB = 7` units, BC = 8 units, AC = 5 units, then the side PQ will be

Options

A. `(sqrt28)/(3)` units
B. `(sqrt88)/(3)` units
C. `(sqrt78)/(3)` units
D. `(sqrt84)/(3)` units

Answer

D

Explanation

`cos C = ((a^(2) + b^(2) -c^(2))/(2ab)) = (1)/(2)`
In `DeltaPQC` we have
`cos C = (CQ^(2) + PC^(2) - PQ^(2))/(2PC.CQ)`
`rArr PQ^(2) = PC^(2) + CQ^(2) - PC.CQ`
`{"where ", PC = (ab)/(b+c) and CQ = (ab)/(a+c)}`
so `PQ = sqrt((100)/(9) + (64)/(9) -(80)/(9)) = (sqrt89)/(3) = (2)/(3) sqrt21`