One side of an equilateral triangle is the line `3x+4y+8=0` and its centroid is at O(1,1). The length of its side is

Options

A. 2
B. `sqrt(5)`
C. `6sqrt(3)`
D. `sqrt(7)`

Answer

To find the length of the side of the equilateral triangle given one side as the line \(3x + 4y + 8 = 0\) and its centroid at \(O(1, 1)\), we can follow these steps:

### Step 1: Find the perpendicular distance from the centroid to the line
The formula for the perpendicular distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:

\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

In our case, the line is \(3x + 4y + 8 = 0\) (where \(A = 3\), \(B = 4\), and \(C = 8\)), and the point is \(O(1, 1)\).

Substituting the values into the formula:

\[ d = \frac{|3(1) + 4(1) + 8|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 4 + 8|}{\sqrt{9 + 16}} = \frac{|15|}{\sqrt{25}} = \frac{15}{5} = 3 \]

### Step 2: Understand the relationship between the centroid and the vertices
In an equilateral triangle, the centroid divides each median in the ratio \(2:1\). If we denote the distance from the centroid \(O\) to the midpoint of the side \(AB\) as \(d\), then the total length of the median \(m\) from vertex \(C\) to the midpoint \(D\) of side \(AB\) is:

\[ m = 3 \times 3 = 9 \]

### Step 3: Relate the median to the side of the triangle
The relationship between the length of a side \(a\) of an equilateral triangle and the length of the median \(m\) is given by the formula:

\[ m = \frac{\sqrt{3}}{2} a \]

Setting \(m = 9\):

\[ 9 = \frac{\sqrt{3}}{2} a \]

### Step 4: Solve for the side length \(a\)
To find \(a\), we rearrange the equation:

\[ a = \frac{9 \times 2}{\sqrt{3}} = \frac{18}{\sqrt{3}} = 6\sqrt{3} \]

Thus, the length of the side of the equilateral triangle is:

\[ \boxed{6\sqrt{3}} \]