Let a, b, c be the side-lengths of a triangle, and l, m,n be the lengths of its medians. Put `K=((l+m+n)/(a+b+c))` Then, as a, b, c vary, K can assume every value in the interval

Options

A. `(1/4,4/5)`
B. `(1/2,4/5)`
C. `(3/4,1)`
D. `(4/5,5/4)`

Answer

C

Explanation

As Ad is medium `rArr ADlt(AB+AC)/2`
`rArrllt(b+c)/2`
Similarly `mlt(c+a)/2" and "nlt(a+b)/2`
`rArr l+m+nlta+b+c`
`rArrl+m+nlta+b+c`
`rArr (l+m+n)/(a+b+c)lt1" (i)"`
Also in the `Delta BGC`
`BG+GCgtBC`
`rArr2/3(m+n)gta`
Similarly `2/3(n+l)gtb`
and `2/3(l+m)gtc`
Hence `4/3(l+m+n)gta+b+c`
`(l+m+n)/(a+b+c)gt3/4" (ii)"`
By (i) and (ii) `(l+m+n)/(a+b+c)in(3/4,1)`