Incentre of the triangle whose vertices are (6, 0), (0, 6) and (7, 7), is

Options

A. `((9)/(2) , (9)/(2))`
B. `((7)/(2) , (7)/(2))`
C. `((11)/(2) , (11)/(2))`
D. none of these

Answer

To find the incenter of the triangle with vertices \( A(6, 0) \), \( B(0, 6) \), and \( C(7, 7) \), we will follow these steps:

### Step 1: Calculate the lengths of the sides of the triangle
We will use the distance formula to find the lengths of the sides \( a \), \( b \), and \( c \).

- **Length of side \( a \) (BC)**:
\[ a = \sqrt{(7 - 0)^2 + (7 - 6)^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \]

- **Length of side \( b \) (AC)**:
\[ b = \sqrt{(7 - 6)^2 + (7 - 0)^2} = \sqrt{1^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \]

- **Length of side \( c \) (AB)**:
\[ c = \sqrt{(6 - 0)^2 + (0 - 6)^2} = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \]

### Step 2: Use the incenter formula
The coordinates of the incenter \( I \) can be calculated using the formula:
\[ I_x = \frac{a x_1 + b x_2 + c x_3}{a + b + c} \]
\[ I_y = \frac{a y_1 + b y_2 + c y_3}{a + b + c} \]
where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices \( A \), \( B \), and \( C \).

Substituting the values:
- \( x_1 = 6, y_1 = 0 \)
- \( x_2 = 0, y_2 = 6 \)
- \( x_3 = 7, y_3 = 7 \)

### Step 3: Calculate \( I_x \)
\[ I_x = \frac{(5\sqrt{2})(6) + (5\sqrt{2})(0) + (6\sqrt{2})(7)}{5\sqrt{2} + 5\sqrt{2} + 6\sqrt{2}} \]
\[ = \frac{30\sqrt{2} + 0 + 42\sqrt{2}}{16\sqrt{2}} = \frac{72\sqrt{2}}{16\sqrt{2}} = \frac{72}{16} = \frac{9}{2} \]

### Step 4: Calculate \( I_y \)
\[ I_y = \frac{(5\sqrt{2})(0) + (5\sqrt{2})(6) + (6\sqrt{2})(7)}{5\sqrt{2} + 5\sqrt{2} + 6\sqrt{2}} \]
\[ = \frac{0 + 30\sqrt{2} + 42\sqrt{2}}{16\sqrt{2}} = \frac{72\sqrt{2}}{16\sqrt{2}} = \frac{72}{16} = \frac{9}{2} \]

### Step 5: Conclusion
Thus, the coordinates of the incenter \( I \) are:
\[ I\left(\frac{9}{2}, \frac{9}{2}\right) \]