Question : The centroid of an equilateral triangle PQR is L. If PQ = 6 cm, the length of PL is :
Option 1: $4 \sqrt{3} \mathrm{~cm}$
Option 2: $3\sqrt{3} \mathrm{~cm}$
Option 3: $2\sqrt{3} \mathrm{~cm}$
Option 4: $5\sqrt{3} \mathrm{~cm}$
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Correct Answer: $2\sqrt{3} \mathrm{~cm}$
Solution :
PQ = 6 cm
The centroid divides the median in the ratio 2 : 1.
Since all the centres lie at the same point L, PL is also the circumradius of Equilateral Triangle PQR.
We know, Circumradius of an equilateral triangle = $\frac{\text{Side of an equilateral triangle}}{\sqrt3}$
⇒ $PL = \frac{PQ}{\sqrt3}$
⇒ $PL = \frac{6}{\sqrt3}$
⇒ $PL = 2\sqrt3\ \text{cm}$
Hence, the correct answer is $2\sqrt3\ \text{cm}$.
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