Multiple choice

In a $\triangle ABC, AB = 10\ cm, BC = 12\ cm$ and $AC = 14\ cm$. Find the length of median $AD$. If $G$ is the centroid, find length of $GA$.

  1. $\dfrac {5}{3}\sqrt {7}, \dfrac {5}{9}\sqrt {7}$
  2. $5\sqrt {7}, 4\sqrt {7}$
  3. $\dfrac {10}{\sqrt {3}}, \dfrac {8}{3}\sqrt {7}$
  4. $4\sqrt {7}, \dfrac {8}{3}\sqrt {7}$
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D Correct answer
Explanation

Using Apollonius theorem for median AD: AB^2 + AC^2 = 2(AD^2 + BD^2). 100 + 196 = 2(AD^2 + 36). 296 = 2(AD^2 + 36), 148 = AD^2 + 36, AD^2 = 112, AD = sqrt(112) = 4*sqrt(7). The centroid G divides the median in 2:1 ratio, so GA = 2/3 * AD = 2/3 * 4*sqrt(7) = 8/3*sqrt(7).

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