Which of the following sets of side lengths could represent the side lengths of a $30$-$60$-$90$ triangle?

• A. $6$, $6\sqrt{3}$, $12$
• B. $4$, $4\sqrt{3}$, $8$
• C. $5$, $5\sqrt{3}$, $10$
• D. $3$, $3\sqrt{3}$, $6$

Answer: (A)

Verified step by step guidance

1. Recall the side length ratios for a 30°-60°-90° triangle: the sides are in the ratio \$1 : \sqrt{3} : 2\(, where the side opposite 30° is the shortest, the side opposite 60° is \)\sqrt{3}$ times the shortest side, and the hypotenuse (opposite 90°) is twice the shortest side.
2. For each given set of side lengths, identify the shortest side, the side that should correspond to \(\sqrt{3}\) times the shortest side, and the longest side (hypotenuse).
3. Check if the middle side is approximately \(\sqrt{3}\) times the shortest side by verifying if \(\text{middle side} = \text{shortest side} \times \sqrt{3}\) holds true.
4. Check if the longest side is twice the shortest side by verifying if \(\text{longest side} = 2 \times \text{shortest side}\) holds true.
5. The set of side lengths that satisfies both conditions correctly represents the sides of a 30°-60°-90° triangle.