The median AD of triangle ABC is bisected at F, and BF is produced to meet the side AC in P. If `AP=lambda AC` then what is the value of `lambda`

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `(2)/(3)`
D. `(1)/(3)`

Answer

D

Explanation

Given, AD is the median of `Delta`ABC. F is mid-point of AF and BF is produced to meet the side AC in P.

Draw DQ||FP
In `Delta`ADQ, F is mid-point of AD and FP||DQ.
`therefore` P is mid-point of AQ (converse of mid-point therorem
`rArr AP = PQ" "...(1)`
In `Delta`BCP, D is mid-point of BC and DQ||BP.
`therefore` Q is midpoint of PC.
`rArr PQ = QC" "...(2)`
From (1), (2) we get, AP = PQ = QC
From figure, AP + PQ + QC = AC
`rArr AP + AP + AP = AC`
`rArr 3 xx AP = AC`
`rArr AP = (1)/(3) xx AC" "therefore" "lambda=(1)/(3)`