The centroid of an equilateral triangle `Delta PQR` is L. If PQ = 18 cm, then the length of PL is :

Options

A. `4sqrt(3) cm`
B. `6sqrt(3) cm`
C. `3sqrt(3) cm`
D. `5 sqrt(3) cm`

Answer

To find the length of PL in the equilateral triangle PQR, we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
In an equilateral triangle, all sides are equal, and the centroid (L) divides each median in a 2:1 ratio. The centroid is also the circumcenter and orthocenter in an equilateral triangle.

### Step 2: Identify the given information
We know that the length of side PQ = 18 cm.

### Step 3: Use the formula for the circumradius of an equilateral triangle
The circumradius (R) of an equilateral triangle can be calculated using the formula:
\[ R = \frac{a}{\sqrt{3}} \]
where \( a \) is the length of a side of the triangle.

### Step 4: Substitute the value of PQ into the formula
Here, \( a = PQ = 18 \) cm. So we substitute this into the formula:
\[ R = \frac{18}{\sqrt{3}} \]

### Step 5: Simplify the expression
To simplify \( \frac{18}{\sqrt{3}} \):
\[ R = \frac{18 \cdot \sqrt{3}}{3} = 6\sqrt{3} \]

### Step 6: Relate the circumradius to PL
Since PL is the circumradius of the triangle PQR, we have:
\[ PL = R = 6\sqrt{3} \]

### Final Answer
The length of PL is \( 6\sqrt{3} \) cm.

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