An equilateral triangle has one vertex at (-1 , -1) and another vertex at `(-sqrt3 , sqrt3)` The third vertex may lie on

Options

A. `(-sqrt2 , sqrt2)`
B. `(sqrt2 , - sqrt2)`
C. `(1 , 1)`
D. `(1 , -1)`

Answer

To find the third vertex of the equilateral triangle given two vertices at (-1, -1) and (-√3, √3), we can follow these steps:

### Step 1: Identify the vertices
Let the vertices of the triangle be:
- A = (-1, -1)
- B = (-√3, √3)
- C = (x, y) (the third vertex we need to find)

### Step 2: Use the distance formula
Since the triangle is equilateral, the distances between all pairs of vertices must be equal. We can use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Calculate the distance AB
Using the distance formula, we calculate the distance AB:
\[ AB = \sqrt{((-√3) - (-1))^2 + (√3 - (-1))^2} \]
\[ = \sqrt{((-√3 + 1)^2 + (√3 + 1)^2)} \]

### Step 4: Calculate the distance AC
Now, we calculate the distance AC:
\[ AC = \sqrt{(x - (-1))^2 + (y - (-1))^2} \]
\[ = \sqrt{(x + 1)^2 + (y + 1)^2} \]

### Step 5: Calculate the distance BC
Next, calculate the distance BC:
\[ BC = \sqrt{(x - (-√3))^2 + (y - √3)^2} \]
\[ = \sqrt{(x + √3)^2 + (y - √3)^2} \]

### Step 6: Set the distances equal
Since all sides are equal in an equilateral triangle, we set the distances equal:
1. \( AB = AC \)
2. \( AB = BC \)

### Step 7: Solve the equations
From \( AB = AC \):
\[ \sqrt{((-√3 + 1)^2 + (√3 + 1)^2)} = \sqrt{(x + 1)^2 + (y + 1)^2} \]
Squaring both sides will eliminate the square root.

From \( AB = BC \):
\[ \sqrt{((-√3 + 1)^2 + (√3 + 1)^2)} = \sqrt{(x + √3)^2 + (y - √3)^2} \]
Again, squaring both sides will eliminate the square root.

### Step 8: Simplify and solve for x and y
After squaring and simplifying both equations, you will end up with a system of equations in terms of x and y. Solve these equations simultaneously to find the coordinates of the third vertex C.

### Step 9: Verify the solution
Once you have the values of x and y, substitute them back into the distance equations to verify that all sides are indeed equal.