Question : Let G be the centroid of the equilateral triangle ABC of perimeter 24 cm. Then the length of AG is:
Option 1: $2 \sqrt 3$
Option 2: $\frac{8}{\sqrt3}$
Option 3: $8 \sqrt 3$
Option 4: $4 \sqrt 3$
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Correct Answer: $\frac{8}{\sqrt3}$
Solution :
The perimeter of the equilateral triangle, P = 24 cm
Side, a = $\frac{P}{3}$ = $\frac{24}{3}$ = 8 cm
Height of the equilateral triangle = $\frac{\sqrt{3}×a}{2}$ = $\frac{\sqrt{3}×8}{2}$ = $4\sqrt3$
As the median is divided in the ratio 2 : 1 at the centroid,
AG = $\frac{2×4\sqrt3}{3}$ = $\frac{8}{\sqrt3}$cm
Hence, the correct answer is $\frac{8}{\sqrt3}$.
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