If ABC is a triangle such that ` A= (1,2) and B = (5,5)` with `BC =9 and AC=12` units, if slope of altitude CD is (D is point on AB) is m/n, then minimum vlaue of `m^(2) + n` is `"_______"`

Answer

To solve the problem, we need to find the minimum value of \( m^2 + n \) given the conditions of triangle ABC with points A and B, and the lengths of sides AC and BC.

### Step-by-Step Solution:

1. **Identify Points A and B**:
- Point A is given as \( A(1, 2) \).
- Point B is given as \( B(5, 5) \).

2. **Calculate the Slope of Line AB**:
- The slope \( m_{AB} \) of line AB can be calculated using the formula:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{5 - 1} = \frac{3}{4} \]

3. **Determine the Slope of Altitude CD**:
- Since CD is the altitude from point C to line AB, the slope of CD, denoted as \( m_{CD} \), must satisfy:
\[ m_{AB} \cdot m_{CD} = -1 \]
- Therefore, we have:
\[ \frac{3}{4} \cdot m_{CD} = -1 \implies m_{CD} = -\frac{4}{3} \]

4. **Express the Slope in Terms of m and n**:
- We can express the slope of CD as \( \frac{m}{n} \). From our previous calculation, we have:
\[ \frac{m}{n} = -\frac{4}{3} \]
- This implies \( m = -4 \) and \( n = 3 \).

5. **Calculate \( m^2 + n \)**:
- Now we need to find the minimum value of \( m^2 + n \):
\[ m^2 + n = (-4)^2 + 3 = 16 + 3 = 19 \]

6. **Final Result**:
- Thus, the minimum value of \( m^2 + n \) is:
\[ \boxed{19} \]