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The question is asking about the ratio between the lengths of the two legs of a 30-60-90 right triangle.

In a 30-60-90 triangle, the sides are in the ratio:
- The shortest leg (opposite the 30° angle) is \( x \),
- The longer leg (opposite the 60° angle) is \( x\sqrt{3} \),
- The hypotenuse is \( 2x \).

Thus, the ratio of the shorter leg to the longer leg is \( 1:\sqrt{3} \).

Now, let's look at the options:
- **A. \( 1:\sqrt{3} \)** – This is correct. It matches the ratio of the two legs.
- **B. \( \sqrt{5}:\sqrt{5} \)** – This simplifies to 1:1, which does not match the 30-60-90 triangle's leg ratio.
- **C. \( \sqrt{2}:\sqrt{2} \)** – This also simplifies to 1:1, so it's incorrect.
- **D. \( 1:\sqrt{5} \)** – This is incorrect since the ratio should be \( 1:\sqrt{3} \), not \( 1:\sqrt{5} \).
- **E. \( \sqrt{2}:\sqrt{3} \)** – This is incorrect because the shorter leg should be 1 (or a multiple of it), not \( \sqrt{2} \).
- **F. \( 2\sqrt{5}:6 \)** – This simplifies to \( \sqrt{5}:3 \), which is also incorrect.

The correct answer is **A**.

Do you need any further details or explanations? Here are some questions you might consider:

1. What are the side ratios for other types of special triangles, such as 45-45-90 triangles?
2. Why is the ratio of the two legs in a 30-60-90 triangle always \( 1:\sqrt{3} \)?
3. How can you derive the formula for the side lengths of a 30-60-90 triangle?
4. What real-world applications use 30-60-90 triangles?
5. How would the ratio change if the triangle were scaled up or down?

**Tip**: Memorizing the side ratios for special right triangles like 30-60-90 and 45-45-90 triangles can be very useful for solving geometry problems quickly!

Which of the following could be the ratio between the lengths of the two legs of a 30-60-90 triangle?

Find out which ratios can represent the lengths of the two legs in a 30-60-90 right triangle. Understand the properties of special right triangles and learn why the correct ratio for a 30-60-90 triangle is 1:√3. Suitable for students in Grades 9-10.

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