Final answer:
The set of values that could be the side lengths of a 30-60-90 triangle is {5, 5\(\sqrt{3}\), 10}, which adheres to the specific side length ratio characteristic of a 30-60-90 triangle.The correct answer is Option A.
Step-by-step explanation:
The student is asking which set of values could be the side lengths of a 30-60-90 triangle. In a 30-60-90 triangle, the lengths follow a specific ratio: The length of the hypotenuse is twice that of the shorter leg, and the length of the longer leg is the shorter leg multiplied by \(\sqrt{3}\). Looking at the given options, we can apply this rule to see which one is correct.
- Option A: \{5, 5\sqrt{3}, 10\} - This option fits the ratio because the hypotenuse (10) is twice the shorter leg (5), and the longer leg (5\sqrt{3}) is the shorter leg (5) multiplied by \(\sqrt{3}\).
- Option B: \{5, 5\sqrt{2}, 10\} - This does not fit the ratio for a 30-60-90 triangle.
- Option C: \{5, 10, 10\sqrt{2}\} - This does not fit the ratio because the second value should be 5\sqrt{3} if the first value is 5 for a 30-60-90 triangle.
- Option D: \{5, 10, 10\sqrt{3}\} - This does not fit the ratio for a 30-60-90 triangle as the hypotenuse should be twice the shorter leg and not the same as the longer leg.
Therefore, the correct answer is Option A.