The points `A(1,-1),B(sqrt(3),sqrt(3))` and `C(0,sqrt(3)-1)` are the vertices of a triangle which is

Options

A. equilateral
B. isosceles
C. right angled
D. obtuse angled

Answer

To determine the nature of the triangle formed by the points A(1, -1), B(√3, √3), and C(0, √3 - 1), we will calculate the lengths of the sides of the triangle and analyze them.

### 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(√3, √3):
\[ AB = \sqrt{(√3 - 1)^2 + (√3 - (-1))^2} \]
Calculating this:
\[ AB = \sqrt{(√3 - 1)^2 + (√3 + 1)^2} \]
\[ = \sqrt{(3 - 2√3 + 1) + (3 + 2√3 + 1)} \]
\[ = \sqrt{(4 - 2√3) + (4 + 2√3)} = \sqrt{8} = 2\sqrt{2} \]

### Step 2: Calculate the length of side BC
For points B(√3, √3) and C(0, √3 - 1):
\[ BC = \sqrt{(0 - √3)^2 + ((√3 - 1) - √3)^2} \]
Calculating this:
\[ BC = \sqrt{(−√3)^2 + (−1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 3: Calculate the length of side AC
For points A(1, -1) and C(0, √3 - 1):
\[ AC = \sqrt{(0 - 1)^2 + ((√3 - 1) - (-1))^2} \]
Calculating this:
\[ AC = \sqrt{(−1)^2 + (√3 - 1 + 1)^2} = \sqrt{1 + (√3)^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 4: Analyze the triangle
Now we have:
- \( AB = 2\sqrt{2} \)
- \( BC = 2 \)
- \( AC = 2 \)

### Step 5: Check for isosceles triangle
Since \( BC = AC \), the triangle is isosceles.

### Step 6: Check for right angle triangle
To check if the triangle is a right triangle, we can use the Pythagorean theorem:
\[ AC^2 + BC^2 = AB^2 \]
Calculating:
\[ (2)^2 + (2)^2 = (2\sqrt{2})^2 \]
\[ 4 + 4 = 8 \]
Since the equation holds true, the triangle is also a right triangle.

### Conclusion
The triangle formed by points A, B, and C is an isosceles right triangle.

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