A triangle has sides of length $48$, $55$, and $73$.  Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle.  Find the value of $a+b$.
1. **Identify the type of triangle**: Given the sides of the triangle are \(48\), \(55\), and \(73\). We need to check if this is a right triangle. For a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the largest side), it is a right triangle if \(a^2 + b^2 = c^2\).

   \[
   48^2 + 55^2 = 2304 + 3025 = 5329
   \]
   \[
   73^2 = 5329
   \]

   Since \(48^2 + 55^2 = 73^2\), the triangle is a right triangle with the hypotenuse \(73\).

2. **Calculate the area of the triangle**: The area \(A\) of a right triangle can be calculated using the legs \(a\) and \(b\):

   \[
   A = \frac{1}{2} \times 48 \times 55 = \frac{1}{2} \times 2640 = 1320
   \]

3. **Find the altitude to the hypotenuse**: The altitude \(h\) to the hypotenuse in a right triangle can be found using the area formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\). Here, the base is the hypotenuse \(73\):

   \[
   1320 = \frac{1}{2} \times 73 \times h
   \]
   \[
   2640 = 73h
   \]
   \[
   h = \frac{2640}{73}
   \]

4. **Simplify the fraction**: The fraction \(\frac{2640}{73}\) is already in its simplest form because \(2640\) and \(73\) are relatively prime (they have no common factors other than 1).

5. **Identify \(a\) and \(b\)**: Here, \(a = 2640\) and \(b = 73\).

6. **Calculate \(a + b\)**:

   \[
   a + b = 2640 + 73 = 2713
   \]

The final answer is \(\boxed{2713}\)