Question

Sir
I am Shaun job of class 9. I have a doubt regarding mathematics on the topic of triangles.This is the question
In any triangle ,the sum of medians is greater than what times the perimeter.
Thank you for your time
Shaun

Answer 1

The answer is 3/4.

proof
''''''''''''''
Let us consider triangle ABC, with centroid G. Let ma, mb and mc be the medians from A, B and C respectively. Then, for instance, AG = 2/3.ma, since G divides the medians in ratio 2:1.
We can apply the triangle inequality to triangles ABG, AGB and GBC, to find:
2/3.ma + 2/3.mb > c (= AB)
2/3.mb + 2/3.mc > a (= BC)
2/3.mc + 2/3.ma > b (= AC)
Adding these three inequalities we find:
4/3.(ma + mb + mc) > a + b + c
ma + mb + mc > 3/4.(a + b + c)