Question : $\triangle ABC$ is an equilateral triangle with a side of 12 cm and AD is the median. Find the length of GD if G is the centroid of $\triangle ABC$.
Option 1: $6 \sqrt{3}$ cm
Option 2: $3 \sqrt{3} $ cm
Option 3: $4 \sqrt{3} $ cm
Option 4: $2 \sqrt{3}$ cm
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Correct Answer: $2 \sqrt{3}$ cm
Solution :
Given: $\triangle ABC$ is an equilateral triangle with a side of 12 cm and AD is the median.
Use the formula,
The height of the equilateral triangle = $\frac{\sqrt3}{2}\times x$ where $x$ is its side.
Also, the centroid divides the median in the ratio of 2 : 1.
The height of the equilateral triangle = $\frac{\sqrt3}{2}\times 12$.
⇒ AD = $6\sqrt3$ cm
Also, AG : GD = 2 : 1
Let AG be $2x$.
⇒ GD = $x$, AD = $3x$
The length of GD is given as,
⇒ $3x=6\sqrt 3$
⇒ $x=2\sqrt 3$ cm
⇒$GD=2\sqrt 3$ cm
Hence, the correct answer is $2 \sqrt{3}$ cm.
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