In a `triangle ABC` the sides `BC=5, CA=4` and `AB=3`. If `A(0,0)` and the internal bisector of angle A meets BC in D `(12/7,12/7)` then incenter of `triangle ABC` is

Options

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

Explanation

To find the incenter of triangle ABC with given sides and coordinates, we can follow these steps:

### Step 1: Identify the coordinates of points B and C
Given:
- A(0,0)
- AB = 3, so B(3,0)
- AC = 4, so C(0,4)

### Step 2: Confirm the coordinates of point D
Point D is given as (12/7, 12/7). This point lies on the internal angle bisector of angle A.

### Step 3: Use the formula for the incenter
The incenter (I) of triangle ABC can be calculated using the formula:
\[ I_x = \frac{aX_A + bX_B + cX_C}{a + b + c} \]
\[ I_y = \frac{aY_A + bY_B + cY_C}{a + b + c} \]
where:
- \(a = BC\)
- \(b = AC\)
- \(c = AB\)
- \(X_A, Y_A\) are the coordinates of point A
- \(X_B, Y_B\) are the coordinates of point B
- \(X_C, Y_C\) are the coordinates of point C

### Step 4: Substitute the values into the formula
From the problem:
- \(a = BC = 5\)
- \(b = AC = 4\)
- \(c = AB = 3\)

Coordinates:
- \(X_A = 0, Y_A = 0\)
- \(X_B = 3, Y_B = 0\)
- \(X_C = 0, Y_C = 4\)

Now substitute these values into the formula for \(I_x\) and \(I_y\):

\[ I_x = \frac{5 \cdot 0 + 4 \cdot 3 + 3 \cdot 0}{5 + 4 + 3} = \frac{12}{12} = 1 \]

\[ I_y = \frac{5 \cdot 0 + 4 \cdot 0 + 3 \cdot 4}{5 + 4 + 3} = \frac{12}{12} = 1 \]

### Step 5: Conclusion
Thus, the coordinates of the incenter \(I\) of triangle ABC is:
\[ I(1, 1) \]