Question : $G$ is the centroid of the equilateral triangle $ABC$. If $AB$ is $9\text{ cm}$, then $AG$ is equal to:
Option 1: $3 \sqrt 3\text{ cm}$
Option 2: $3\text{ cm}$
Option 3: $\frac{3 \sqrt 3}{2}\text{ cm}$
Option 4: $6 \text{ cm}$
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Correct Answer: $3 \sqrt 3\text{ cm}$
Solution :
The side of an equilateral triangle ($a$) $=9\text{ cm}$
Median $AD$ of equilateral triangle $=\frac{\sqrt{3}}{2}a$
⇒ ${AD} = \frac{\sqrt{3}}{2}\times 9$
Also, a centroid divides the median in the ratio of $2:1$.
So, $ AG : GD = 2 : 1$
⇒ $\frac{AG}{AD}=\frac{2}{3}$
⇒ $AG = \frac{2}{3} \times AD$
⇒ $AG= \frac{2}{3} \times \frac{\sqrt{3}}{2}\times 9 = 3\sqrt{3}$
Hence, the correct answer is $3\sqrt{3}\text{ cm}$.
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