The points `(1,1),(-1,-1)` and `(-sqrt(3),sqrt(3))` are the angular points of a triangle, then the triangle is

Options

A. right angled
B. isoceles
C. equilateral
D. None

Answer

To determine the type of triangle formed by the points \( A(1, 1) \), \( B(-1, -1) \), and \( C(-\sqrt{3}, \sqrt{3}) \), we will calculate the lengths of the sides of the triangle using the distance formula.

### Step 1: Calculate the length of side AB

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For points \( A(1, 1) \) and \( B(-1, -1) \):

\[ AB = \sqrt{((-1) - 1)^2 + ((-1) - 1)^2} \]
\[ = \sqrt{(-2)^2 + (-2)^2} \]
\[ = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

### Step 2: Calculate the length of side BC

For points \( B(-1, -1) \) and \( C(-\sqrt{3}, \sqrt{3}) \):

\[ BC = \sqrt{((- \sqrt{3}) - (-1))^2 + (\sqrt{3} - (-1))^2} \]
\[ = \sqrt{(-\sqrt{3} + 1)^2 + (\sqrt{3} + 1)^2} \]
\[ = \sqrt{(1 - \sqrt{3})^2 + (1 + \sqrt{3})^2} \]
Calculating \( (1 - \sqrt{3})^2 \) and \( (1 + \sqrt{3})^2 \):

\[ (1 - \sqrt{3})^2 = 1 - 2\sqrt{3} + 3 = 4 - 2\sqrt{3} \]
\[ (1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3} \]

Now, adding these:

\[ BC = \sqrt{(4 - 2\sqrt{3}) + (4 + 2\sqrt{3})} = \sqrt{8} = 2\sqrt{2} \]

### Step 3: Calculate the length of side CA

For points \( C(-\sqrt{3}, \sqrt{3}) \) and \( A(1, 1) \):

\[ CA = \sqrt{(1 - (-\sqrt{3}))^2 + (1 - \sqrt{3})^2} \]
\[ = \sqrt{(1 + \sqrt{3})^2 + (1 - \sqrt{3})^2} \]

Calculating \( (1 + \sqrt{3})^2 \) and \( (1 - \sqrt{3})^2 \):

\[ (1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3} \]
\[ (1 - \sqrt{3})^2 = 1 - 2\sqrt{3} + 3 = 4 - 2\sqrt{3} \]

Now, adding these:

\[ CA = \sqrt{(4 + 2\sqrt{3}) + (4 - 2\sqrt{3})} = \sqrt{8} = 2\sqrt{2} \]

### Step 4: Compare the lengths of the sides

We have:

- \( AB = 2\sqrt{2} \)
- \( BC = 2\sqrt{2} \)
- \( CA = 2\sqrt{2} \)

Since all three sides are equal, we conclude that the triangle is an **equilateral triangle**.

### Final Conclusion

The triangle formed by the points \( (1, 1) \), \( (-1, -1) \), and \( (-\sqrt{3}, \sqrt{3}) \) is an **equilateral triangle**.

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