Calculate Triangle ABC Area: Using 12-Unit Rectangle and 13-Unit Diagonal

Let's solve this step-by-step:

• Step 1: Identify the given information.
  We know the rectangle $ABCD$, is divided by its diagonal $AC$. The length $AB$ is $12$, and the diagonal $AC$ is $13$.
• Step 2: Apply Pythagorean theorem to find $BC$, which acts as the height.
  Using the Pythagorean theorem in $\triangle ABC$ gives us: $AC = \sqrt{AB^2 + BC^2}$ Given $AC = 13$ and $AB = 12$, we set up the equation: $13 = \sqrt{12^2 + BC^2}$ Squaring both sides leads to: $169 = 144 + BC^2$ $BC^2 = 169 - 144 = 25$ Thus, $BC = \sqrt{25} = 5$.
• Step 3: Calculate the area of $\triangle ABC$.
  The area can be found using the formula: $\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC$ $= \frac{1}{2} \times 12 \times 5$ $= \frac{1}{2} \times 60 = 30$

Therefore, the area of triangle $ABC$ is $\boxed{30}$.