if a right triangle has a hypotenuse  of 9cm and the smaller angles of the triangle measure 30 degrees and 60  degrees what will be the length of the leg opposite of the 30 degree angle

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.

Learn how to calculate the length of the leg opposite the 30-degree angle in a right triangle with a hypotenuse of 9 cm and angles measuring 30 degrees, 60 degrees, and 90 degrees. Follow a step-by-step solution with clear explanations and calculations.

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