Final answer:
Only options A (3:3√3) and C (1:√3) are correct ratios for the lengths of the legs of a 30-60-90 triangle, as they reflect the standard ratio of 1:√3 for this type of right triangle.
Step-by-step explanation:
In a 30-60-90 triangle, the ratio of the lengths of the shorter leg to the longer leg is always 1:√3, and this ratio is derived from the properties of this special right triangle. In such a triangle, the hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg. Therefore, our valid options from the ones given would be ratios that reflect this relationship.
- Option A, 3:3√3, simplifies to 1:√3, which is a correct ratio.
- Option C, 1:√3, is explicitly the correct ratio for a 30-60-90 triangle.
Options B, D, E, and F are not correct ratios for a 30-60-90 triangle because they do not reflect the established relationship between the legs of such a triangle.