Right Triangle Area: Calculate Using Base 12 and Height 9

To solve this problem, we'll follow these steps:

• Step 1: Identify the base and height from the given right triangle.
• Step 2: Apply the area formula for the triangle.
• Step 3: Calculate the area using the given side lengths.

Now, let's work through each step:
Step 1: In triangle $\triangle ABC$, the sides $AB$ and $BC$ are the legs, where $AB = 9$ and $BC = 12$. These serve as the base and height.

Step 2: We'll use the formula for the area of a right triangle:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.

Step 3: Plugging the values into the formula gives:
$\text{Area} = \frac{1}{2} \times 9 \times 12$.

Perform the calculation:
$\text{Area} = \frac{1}{2} \times 108 = 54$.

Therefore, the area of triangle $\triangle ABC$ is $\text{Area} = 54$.

Thus, the correct answer is $\boxed{54}$, which corresponds to choice $\#3$.