The incetre of the triangle with vertices `(1,sqrt(3)),(0,0)` and (2,0) is

Options

A. `(1,(sqrt(3))/(2))`
B. `((2)/(3),(1)/(sqrt(3)))`
C. `((2)/(3),(sqrt(3))/(2))`
D. `(1,(1)/(sqrt(3)))`

Answer

To find the incenter of the triangle with vertices \( A(1, \sqrt{3}), B(0, 0), C(2, 0) \), we will follow these steps:

### Step 1: Identify the vertices
The vertices of the triangle are:
- \( A(1, \sqrt{3}) \)
- \( B(0, 0) \)
- \( C(2, 0) \)

### Step 2: Calculate the lengths of the sides
We need to calculate the lengths of the sides of the triangle using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

1. Length \( a = BC \):
\[ a = \sqrt{(2 - 0)^2 + (0 - 0)^2} = \sqrt{2^2} = 2 \]

2. Length \( b = AC \):
\[ b = \sqrt{(2 - 1)^2 + (0 - \sqrt{3})^2} = \sqrt{(1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

3. Length \( c = AB \):
\[ c = \sqrt{(1 - 0)^2 + (\sqrt{3} - 0)^2} = \sqrt{(1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 3: Use the incenter formula
The coordinates of the incenter \( (I_x, I_y) \) can be calculated using the formula:
\[ I_x = \frac{a x_1 + b x_2 + c x_3}{a + b + c} \]
\[ I_y = \frac{a y_1 + b y_2 + c y_3}{a + b + c} \]

Substituting the values:
- \( a = 2 \), \( b = 2 \), \( c = 2 \)
- \( (x_1, y_1) = (1, \sqrt{3}) \)
- \( (x_2, y_2) = (0, 0) \)
- \( (x_3, y_3) = (2, 0) \)

### Step 4: Calculate \( I_x \)
\[ I_x = \frac{2 \cdot 1 + 2 \cdot 0 + 2 \cdot 2}{2 + 2 + 2} = \frac{2 + 0 + 4}{6} = \frac{6}{6} = 1 \]

### Step 5: Calculate \( I_y \)
\[ I_y = \frac{2 \cdot \sqrt{3} + 2 \cdot 0 + 2 \cdot 0}{2 + 2 + 2} = \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} \]

### Step 6: Conclusion
The coordinates of the incenter are:
\[ I(1, \frac{\sqrt{3}}{3}) \]