Question : In $\triangle A B C,\ D$ is the mid-point of $BC$ and $G$ is the centroid. If $G D = 10\ \text{cm} $, the length of $AD$ is_____.
Option 1: 20 cm
Option 2: 30 cm
Option 3: 15 cm
Option 4: 10 cm
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Correct Answer: 30 cm
Solution : The centroid divides each median into two segments whose lengths are in the ratio of 2 : 1. $GD$ = 10 cm Let the length of $AG$ be $x$. $\therefore \frac{2}{1} = \frac{x}{10}$ $\Rightarrow x = 20\ \text{cm}$ Now, $AD = GD + AG = 10 + 20 = 30$ cm Hence, the correct answer is 30 cm.
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