If the distance from the vertex to the centroid of an equilateral triangle is 6 cm, then what is the area of the triangle?

Options

A. `24 cm^(2)`
B. `27 sqrt(3) cm^(2)`
C. `12 cm^(2)`
D. `12sqrt(3)cm^(2)`

Answer

To find the area of the equilateral triangle given that the distance from a vertex to the centroid is 6 cm, we can follow these steps:

### Step 1: Understanding the relationship between the centroid and the vertex
In an equilateral triangle, the centroid divides each median in a 2:1 ratio. This means that if we denote the distance from a vertex (let's say vertex A) to the centroid (G) as AG, and the distance from the centroid to the midpoint of the opposite side (let's say D) as GD, we have:

AG : GD = 2 : 1

### Step 2: Finding the length of the median
Given that AG = 6 cm, we can find GD using the ratio:

If AG = 6 cm, then GD can be calculated as follows:
- Since AG : GD = 2 : 1, we can express GD as:
\[ GD = \frac{1}{2} \times AG = \frac{1}{2} \times 6 = 3 \text{ cm} \]

### Step 3: Finding the total length of the median
The total length of the median AD (which is the sum of AG and GD) can be calculated as:
\[ AD = AG + GD = 6 \text{ cm} + 3 \text{ cm} = 9 \text{ cm} \]

### Step 4: Relating the median to the side length of the triangle
For an equilateral triangle, the length of the median (m) can be related to the side length (s) using the formula:
\[ m = \frac{\sqrt{3}}{2} s \]
We have found that the median AD = 9 cm, so we can set up the equation:
\[ \frac{\sqrt{3}}{2} s = 9 \]

### Step 5: Solving for the side length (s)
To find the side length s, we rearrange the equation:
\[ s = \frac{9 \times 2}{\sqrt{3}} = \frac{18}{\sqrt{3}} = 6\sqrt{3} \text{ cm} \]

### Step 6: Finding the area of the triangle
The area (A) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting the value of s:
\[ A = \frac{\sqrt{3}}{4} (6\sqrt{3})^2 = \frac{\sqrt{3}}{4} \times 108 = \frac{108\sqrt{3}}{4} = 27\sqrt{3} \text{ cm}^2 \]

### Final Answer
The area of the equilateral triangle is \( 27\sqrt{3} \text{ cm}^2 \).

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