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 \( m^2 + n \) where \( m/n \) is the slope of the altitude \( CD \) from point \( C \) to line segment \( AB \). Let's go through the steps systematically.

### Step 1: Find the coordinates of points A and B
Given:
- \( A = (1, 2) \)
- \( B = (5, 5) \)

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

### Step 3: Determine the slope of the altitude CD
Since \( CD \) is the altitude from point \( C \) to line segment \( AB \), it will be perpendicular to \( AB \). The product of the slopes of two perpendicular lines is \(-1\). Therefore, if the slope of \( AB \) is \( \frac{3}{4} \), the slope \( m_{CD} \) of the altitude \( CD \) is:
\[ m_{CD} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3} \]

### Step 4: Express the slope in terms of \( m \) and \( n \)
From the slope \( m_{CD} = -\frac{4}{3} \), we can identify:
- \( m = -4 \)
- \( n = 3 \)

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

### Conclusion
Thus, the minimum value of \( m^2 + n \) is:
\[ \boxed{19} \]