If the coordinates of the centroid and a vertex oc an equilaterqal triangle are (1,1) and (1,2) respectively, then the coordinates of another vertex, are

Options

A. `((2-sqrt(3))/(2),-(1)/(2))`
B. `((2+3sqrt(3))/(2),-(1)/(2))`
C. `((2+sqrt(3))/(2),-(1)/(2))`
D. none of these

Answer

To find the coordinates of the other vertex of the equilateral triangle given the centroid and one vertex, we can follow these steps:

### Step 1: Understand the properties of the centroid
The centroid (G) of a triangle with vertices A, B, and C is given by the formula:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices A, B, and C respectively.

### Step 2: Assign the known values
Let’s denote:
- The centroid \( G(1, 1) \)
- One vertex \( A(1, 2) \)
- The other two vertices as \( B(x_1, y_1) \) and \( C(x_2, y_2) \)

### Step 3: Set up the equations based on the centroid formula
Using the centroid formula, we can write:
\[ 1 = \frac{1 + x_1 + x_2}{3} \]
\[ 1 = \frac{2 + y_1 + y_2}{3} \]

### Step 4: Solve for \( x_1 + x_2 \) and \( y_1 + y_2 \)
From the first equation:
\[ 3 = 1 + x_1 + x_2 \]
\[ x_1 + x_2 = 2 \quad \text{(Equation 1)} \]

From the second equation:
\[ 3 = 2 + y_1 + y_2 \]
\[ y_1 + y_2 = 1 \quad \text{(Equation 2)} \]

### Step 5: Use the properties of the equilateral triangle
In an equilateral triangle, the distance between any two vertices is equal. Therefore, we can use the distance formula to set up equations for the distances \( AB \), \( AC \), and \( BC \).

1. Distance \( AB \):
\[ AB = \sqrt{(x_1 - 1)^2 + (y_1 - 2)^2} \]

2. Distance \( AC \):
\[ AC = \sqrt{(x_2 - 1)^2 + (y_2 - 2)^2} \]

3. Distance \( BC \):
\[ BC = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]

Since \( AB = AC = BC \), we can set these distances equal to each other.

### Step 6: Set up the equations for distances
From \( AB = AC \):
\[ (x_1 - 1)^2 + (y_1 - 2)^2 = (x_2 - 1)^2 + (y_2 - 2)^2 \]

From \( AB = BC \):
\[ (x_1 - 1)^2 + (y_1 - 2)^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 \]

### Step 7: Solve the equations
Substituting \( y_2 = 1 - y_1 \) from Equation 2 into the distance equations, we can solve for \( x_1 \) and \( y_1 \).

### Step 8: Find the coordinates
After solving the equations, we find the coordinates of the other vertex \( B \) and \( C \).

### Final Result
After performing the calculations, we find that the coordinates of the other vertex are:
\[ \left(1 + \frac{\sqrt{3}}{2}, \frac{1}{2}\right) \]