The centroid of an equilateral triangle PQR is L. If $PQ = 6 ext{ cm}$, what is the length of $PL$?

Question

The centroid of an equilateral triangle PQR is L. If $PQ = 6 	ext{ cm}$, what is the length of $PL$?

Accepted Answer

**Solution:**

1. In an equilateral triangle, all centers (centroid, circumcenter, incentre, orthocenter) coincide at the same point.

2. The circumradius $R$ of an equilateral triangle with side length $a$ is given by:
   $$R = \frac{a}{\sqrt{3}}$$

3. Given $PQ = 6 	ext{ cm}$, the circumradius (and hence the length $PL$) is:
   $$PL = \frac{6}{\sqrt{3}} = 2\sqrt{3} 	ext{ cm}$$

4. Since $L$ is the centroid and coincides with the circumcenter, $PL$ equals the circumradius.

**Answer:** $2\sqrt{3} 	ext{ cm}$

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