1. The problem asks to determine if the ratio $12 : 12\sqrt{3} : 24$ corresponds to the side lengths of a $45^\circ$-$45^\circ$-$90^\circ$ triangle, a $30^\circ$-$60^\circ$-$90^\circ$ triangle, or neither. 2. Recall the side length ratios for these special right triangles: - For a $45^\circ$-$45^\circ$-$90^\circ$ triangle, the sides are in the ratio $1 : 1 : \sqrt{2}$. - For a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. 3. Let's simplify the given ratio $12 : 12\sqrt{3} : 24$ by dividing all terms by 12: $$\frac{12}{12} : \frac{12\sqrt{3}}{12} : \frac{24}{12} = 1 : \sqrt{3} : 2$$ 4. This simplified ratio exactly matches the $30^\circ$-$60^\circ$-$90^\circ$ triangle side ratio. 5. Therefore, the given ratio corresponds to a $30^\circ$-$60^\circ$-$90^\circ$ triangle, not the other options. **Final answer:** The ratio $12 : 12\sqrt{3} : 24$ is for a $30^\circ$-$60^\circ$-$90^\circ$ triangle.