In the given problem, we have a right triangle $\triangle ABC$, where $\angle B = 90^\circ$, and the altitude from $B$ to $AC$, denoted as $BD$, has a length of 12. The goal is to find the minimum possible length of the hypotenuse $AC$, assuming all side lengths and parameters of the triangle are integers.

Step-by-Step Solution:

1. Using the property of a right triangle: Since $BD$ is the altitude to the hypotenuse $AC$, we can use the following formula that relates the altitude, the sides, and the hypotenuse of a right triangle: $BD = \frac{ab}{c}$ where $a = AB$, $b = BC$, and $c = AC$ is the hypotenuse. We are given that $BD = 12$.

   So, the equation becomes: $\frac{ab}{c} = 12$ or equivalently: $ab = 12c$ This equation will help us find integer values for $a$, $b$, and $c$.

2. Finding integer solutions: The next step is to find integer values of $a$, $b$, and $c$ that satisfy the equation $ab = 12c$. Since we want to minimize $c$, we can try different factorizations of 12.

   • Let $a = 3$ and $b = 4$, then: $ab = 3 \times 4 = 12$ which implies: $12 = 12c \quad \Rightarrow \quad c = 1$ However, this gives a non-realistic value for $c$, so let's try different pairs of $a$ and $b$.

   • Let $a = 6$ and $b = 8$. Then: $ab = 6 \times 8 = 48$ which implies: $48 = 12c \quad \Rightarrow \quad c = 4$ which again leads to inconsistent triangle dimensions.

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