In a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees, there is a special relationship between the sides:

• The side opposite the 30-degree angle is half the length of the hypotenuse.
• The side opposite the 60-degree angle is $\sqrt{3}/2$ times the length of the hypotenuse.

Given:

• Hypotenuse $= 9 \, \text{cm}$

Step 1: Calculate the length of the leg opposite the 30-degree angle

The length of the leg opposite the 30-degree angle is: $\text{Opposite leg} = \frac{1}{2} \times \text{Hypotenuse} = \frac{1}{2} \times 9 \, \text{cm} = 4.5 \, \text{cm}$

Final Answer:

The length of the leg opposite the 30-degree angle is 4.5 cm.

Would you like more details on this solution or have any other questions?

Here are 5 related questions to further your understanding:

1. How do you find the length of the leg opposite the 60-degree angle?
2. What is the relationship between the sides in a 45-45-90 triangle?
3. How would the length of the sides change if the hypotenuse was 12 cm instead of 9 cm?
4. What is the area of the triangle given the hypotenuse and one leg?
5. How can you use trigonometric functions to find the sides of any right triangle?

Tip: In special right triangles, like the 30-60-90 triangle, knowing just one side can help you determine the lengths of the other sides quickly using the known ratios.