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Let's solve the problem by finding the exact values of the side lengths \( b \) and \( a \) in the given right triangles.

### 1. Triangle with Angles 30°, 60°, and 90°

This is a 30-60-90 triangle. In a 30-60-90 triangle, the ratios of the side lengths are:
- The side opposite the 30° angle is \( x \).
- The side opposite the 60° angle is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).

Here, the side opposite the 30° angle is 6, so we can set:
\[
x = 6
\]

Then, the side opposite the 60° angle (which is \( b \)) is:
\[
b = x\sqrt{3} = 6\sqrt{3}
\]

### 2. Triangle with Angles 45°, 45°, and 90°

This is a 45-45-90 triangle. In a 45-45-90 triangle, the ratios of the side lengths are:
- Each leg is \( x \).
- The hypotenuse is \( x\sqrt{2} \).

Here, one leg is given as 8. Therefore, we can set:
\[
x = 8
\]

The hypotenuse (which is \( a \)) will be:
\[
a = x\sqrt{2} = 8\sqrt{2}
\]

### Final Answers

\[
b = 6\sqrt{3}, \quad a = 8\sqrt{2}
\]

Would you like further details, or do you have any questions?

### Additional Questions to Explore

1. How do you find the side lengths of a 45-45-90 triangle if only the hypotenuse is known?
2. What is the general formula for finding side lengths in a 30-60-90 triangle?
3. How can trigonometric functions like sine or cosine be used to verify these results?
4. What are the characteristics of special right triangles?
5. How do you derive the ratios of side lengths for 30-60-90 and 45-45-90 triangles?

**Tip:** Understanding the side length ratios of special triangles is key to solving many geometry problems quickly!

For the right triangles below, find the exact values of the side lengths b and a. If necessary, write your responses in simplified radical form.

Find the exact side lengths of special right triangles, including 30°-60°-90° and 45°-45°-90°, in simplified radical form. Learn how to use side length ratios to solve geometry problems efficiently. Suitable for Grades 9-11.

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