The co-ordinates of the third vertex C on the x-axis so that `A(2,0)`, `B(2+2sqrt3,6)` and C form an equilateral triangle is :

Options

A. (,4)
B. (1,5)
C. (2,3)
D. (5,1)

Answer

To find the coordinates of the third vertex \( C \) on the x-axis such that points \( A(2,0) \), \( B(2 + 2\sqrt{3}, 6) \), and \( C \) form an equilateral triangle, we can follow these steps:

### Step 1: Identify the coordinates of points A and B
- Point \( A \) is given as \( (2, 0) \).
- Point \( B \) is given as \( (2 + 2\sqrt{3}, 6) \).

### Step 2: Calculate the distance AB
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( A \) and \( B \):
\[ AB = \sqrt{((2 + 2\sqrt{3}) - 2)^2 + (6 - 0)^2} \]
\[ = \sqrt{(2\sqrt{3})^2 + 6^2} \]
\[ = \sqrt{4 \cdot 3 + 36} \]
\[ = \sqrt{12 + 36} \]
\[ = \sqrt{48} \]
\[ = 4\sqrt{3} \]

### Step 3: Determine the coordinates of point C
Since \( C \) lies on the x-axis, its coordinates can be represented as \( (x, 0) \).

### Step 4: Set up the condition for an equilateral triangle
For triangle \( ABC \) to be equilateral, all sides must be equal. Thus, we need to set the distances \( AC \) and \( BC \) equal to \( AB \):
1. Distance \( AC \):
\[ AC = \sqrt{(x - 2)^2 + (0 - 0)^2} = |x - 2| \]
2. Distance \( BC \):
\[ BC = \sqrt{(x - (2 + 2\sqrt{3}))^2 + (0 - 6)^2} = \sqrt{(x - (2 + 2\sqrt{3}))^2 + 36} \]

### Step 5: Set the equations equal
Setting \( AC = AB \):
\[ |x - 2| = 4\sqrt{3} \]
This gives us two cases:
1. \( x - 2 = 4\sqrt{3} \)
2. \( x - 2 = -4\sqrt{3} \)

### Step 6: Solve for x
1. For \( x - 2 = 4\sqrt{3} \):
\[ x = 2 + 4\sqrt{3} \]
2. For \( x - 2 = -4\sqrt{3} \):
\[ x = 2 - 4\sqrt{3} \]

### Step 7: Determine which solution is valid
Since \( C \) must be on the x-axis, we need to check the validity of both solutions. The point \( 2 - 4\sqrt{3} \) will be negative (as \( 4\sqrt{3} \approx 6.93 \)), which is not valid in this context. Thus, the only valid coordinate for point \( C \) is:
\[ C(2 + 4\sqrt{3}, 0) \]

### Final Answer
The coordinates of the third vertex \( C \) are \( (2 + 4\sqrt{3}, 0) \).