If two vertices of an equilateral triangle are `(1,1) and (-1,1)` then the third vertex may be

Options

A. `(-sqrt3,-sqrt3)`
B. `(-sqrt3,sqrt3)`
C. `(sqrt3,-sqrt3)`
D. `(sqrt3,sqrt3)`

Answer

To find the third vertex of the equilateral triangle with the given vertices (1, 1) and (-1, 1), we can follow these steps:

### Step 1: Identify the Given Points
Let the vertices of the triangle be:
- A(1, 1)
- B(-1, 1)

### Step 2: Calculate the Length of Side AB
Using the distance formula, we can calculate the length of side AB:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
\[ AB = \sqrt{((-1) - 1)^2 + (1 - 1)^2} = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2 \]

### Step 3: Determine the Length of Each Side
Since the triangle is equilateral, all sides are equal. Therefore, the length of each side, including the sides AC and BC, is also 2.

### Step 4: Set Up the Equation for the Third Vertex
Let the coordinates of the third vertex C be (a, b). We need to ensure that the lengths AC and BC are also equal to 2.

1. For AC:
\[ AC = \sqrt{(a - 1)^2 + (b - 1)^2} = 2 \]
Squaring both sides:
\[ (a - 1)^2 + (b - 1)^2 = 4 \quad \text{(1)} \]

2. For BC:
\[ BC = \sqrt{(a + 1)^2 + (b - 1)^2} = 2 \]
Squaring both sides:
\[ (a + 1)^2 + (b - 1)^2 = 4 \quad \text{(2)} \]

### Step 5: Solve the Equations
Now we have two equations (1) and (2):
1. \((a - 1)^2 + (b - 1)^2 = 4\)
2. \((a + 1)^2 + (b - 1)^2 = 4\)

Subtract equation (1) from equation (2):
\[ (a + 1)^2 - (a - 1)^2 = 0 \]
Expanding both sides:
\[ (a^2 + 2a + 1) - (a^2 - 2a + 1) = 0 \]
This simplifies to:
\[ 4a = 0 \implies a = 0 \]

### Step 6: Substitute a Back into One of the Equations
Now substitute \(a = 0\) into equation (1):
\[ (0 - 1)^2 + (b - 1)^2 = 4 \]
This simplifies to:
\[ 1 + (b - 1)^2 = 4 \]
\[ (b - 1)^2 = 3 \]
Taking the square root:
\[ b - 1 = \pm \sqrt{3} \]
Thus, we have two possible values for \(b\):
\[ b = 1 + \sqrt{3} \quad \text{or} \quad b = 1 - \sqrt{3} \]

### Step 7: Write the Possible Coordinates for Vertex C
Therefore, the coordinates of the third vertex C can be:
\[ C(0, 1 + \sqrt{3}) \quad \text{or} \quad C(0, 1 - \sqrt{3}) \]

### Final Answer
The third vertex of the equilateral triangle may be at the points:
- \(C(0, 1 + \sqrt{3})\)
- \(C(0, 1 - \sqrt{3})\)