Which set of numbers can represent the side lengths of a $30$-$60$-$90$ special right triangle?

• A. $1$, $1$, $\sqrt{2}$
• B. $3$, $4$, $5$
• C. $2$, $2$, $2$
• D. $1$, $\sqrt{3}$, $2$

Answer: (D)

Verified step by step guidance

1. Recall the side length ratios of a 30°-60°-90° triangle: the sides are in the ratio 1 : \(\sqrt{3}\) : 2, where 1 is the length of the side opposite the 30° angle, \(\sqrt{3}\) is opposite the 60° angle, and 2 is the hypotenuse opposite the 90° angle.
2. Examine each given set of numbers and check if they can be scaled to match the ratio 1 : \(\sqrt{3}\) : 2.
3. For the set (1, 1, \(\sqrt{2}\)), note that the sides are not in the ratio 1 : \(\sqrt{3}\) : 2, so this set does not represent a 30°-60°-90° triangle.
4. For the set (3, 4, 5), recognize this is a Pythagorean triple but does not match the 30°-60°-90° ratio, so it is not the correct set.
5. For the set (1, \(\sqrt{3}\), 2), observe that it exactly matches the 30°-60°-90° triangle side ratios, confirming this set can represent the side lengths of such a triangle.