What is the greatest possible perimeter of a right angled triangle with integer side lengths if one of the sides has length 12?

Answer

To find the greatest possible perimeter of a right-angled triangle with integer side lengths, where one of the sides has a length of 12, we can follow these steps:

### Step 1: Identify the Pythagorean Triplet Form
The sides of a right-angled triangle can be represented using the Pythagorean triplet formula:
- \( a = 2m \)
- \( b = m^2 - 1 \)
- \( c = m^2 + 1 \)

where \( m \) is a positive integer.

### Step 2: Assign the Length of 12
Since we want to maximize the perimeter, we will assume that 12 is the smallest side. Therefore, we set:
\[ 2m = 12 \]

### Step 3: Solve for \( m \)
From the equation \( 2m = 12 \), we can solve for \( m \):
\[ m = \frac{12}{2} = 6 \]

### Step 4: Calculate the Other Sides
Now that we have \( m \), we can calculate the other two sides:
- For the side \( b \):
\[ b = m^2 - 1 = 6^2 - 1 = 36 - 1 = 35 \]
- For the side \( c \):
\[ c = m^2 + 1 = 6^2 + 1 = 36 + 1 = 37 \]

### Step 5: Calculate the Perimeter
Now we can find the perimeter \( P \) of the triangle:
\[ P = 12 + 35 + 37 \]

Calculating this gives:
\[ P = 12 + 35 + 37 = 84 \]

### Conclusion
Thus, the greatest possible perimeter of a right-angled triangle with integer side lengths, where one side is 12, is:
\[ \boxed{84} \]

---