If AD BE CF arc the medians of a Delta ABC then the correct relation between the sum of the squares ...........

Quantitative Aptitude ( CCS) Geometry

If AD, BE, CF arc the medians of a ΔABC then the correct relation between the sum of the squares of sides to the sum of the squares of median is

2(AB² + BC² + AC²) = 3(AD² + BE² + CF²)

4(AB² + BC² + AC²) = 3(AD² + BE² + CF²)

3(AB² + BC² + AC²) = 4(AD² + BE² + CF²)

None of the above

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Explanation:

Let G be the centroid of ΔABC.
In ΔABC,
[? The sum of the squares of any two sides is equal to twice the square of the half of the third side together with the square of the median bisecting the third side]

∴ AB² + AC² = 2AD² + 2(1/2BC²) ......i
⇒ AB² + BC² = 2AD² + 1/2BC²
BC² + AB² = 2BE² + 1/2AC² ..........ii
BC² + AC² = 2CF² + 1/2AB²..........iii
Adding (i), (ii) and (iii), we get
2(AB² + BC² + AC²
= 2(AD² + BE² + CF²) + 1/2(AB² + BC² + AC²)
∴ 3(AB² + BC² + AC²) = 4(AD² + BE² + CF²)