Let a,b,c be the side-lengths of a triangle, and l, m, n be the lengths of its medians. Let K=\frac{l+m+n}{a+b+c} Then, as a, b, c vary, K can assume every value in the interval

KVPY SA Exam 2018 Question Paper

Question : 9 of 80

Marks: +1, -0

Let a,b,c be the side-lengths of a triangle, and l, m, n be the lengths of its medians. Let

K={l+m+n}/{a+b+c}

Then, as a, b, c vary, K can assume every value in the interval

Solution:

As AD is medium ⇒

AD<{AB + AC}/2

⇒

l <{b+c}/2

Similarly

m< {c+ a} /2\and n <{a + b}/2

⇒

l+ m +n +a +b +c

⇒

{l+m+n}/{a+b+c}<1

......(i)
Also in the

∆ \BGC

BG + GC > BC
⇒

2/3(m+n)>a

Similarly

2/3(n+l)>b

and

2/3(l+m)>c

Hence

4/3(l+m+n)>a+b+c

{l+ m +n }/{a +b +c}>3/4.....

(ii)
By (i) and (ii)

{l+ m +n }/{a +b +c} ∈(3/4,1)

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