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In the given problem, we are asked to find the side lengths \( h \) and \( b \) for two right triangles, using trigonometric relationships.

### Triangle 1 (with angle 45°):
- We know the hypotenuse is 5, and the triangle has a 45° angle. In a 45°-45°-90° triangle, both legs are equal, and the ratio of the leg to the hypotenuse is \( \frac{1}{\sqrt{2}} \).
  
Using this, we can find \( h \):

\[
h = 5 \times \frac{1}{\sqrt{2}} = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2}
\]

### Triangle 2 (with angle 60° and 30°):
- We know the side opposite the 60° angle is 6, and we need to find \( b \). In a 30°-60°-90° triangle, the sides follow these ratios:
  - The side opposite the 30° angle is \( \frac{1}{2} \) of the hypotenuse.
  - The side opposite the 60° angle (which we know is 6) is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.

Let the hypotenuse be \( b \). Using the ratio \( \frac{\sqrt{3}}{2} b = 6 \), we can solve for \( b \):

\[
b = \frac{6 \times 2}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3}
\]

### Final Answers:
- \( h = \frac{5\sqrt{2}}{2} \)
- \( b = 4\sqrt{3} \)

Would you like a further breakdown or have any questions?

Here are 5 related questions to explore:
1. How do you derive the ratios for 30°-60°-90° triangles?
2. What is the Pythagorean theorem, and how can it verify these answers?
3. Can trigonometric functions be used to solve this problem differently?
4. How would the solution change if the hypotenuse of the 45° triangle was different?
5. What is the significance of simplifying answers in radical form?

**Tip**: Always simplify radicals for clarity and precision when working with geometry.

For the right triangles below, find the exact values of the side lengths h and b.

Learn how to find the exact values of the side lengths h and b in right triangles using special right triangle ratios. This problem involves understanding the 45°-45°-90° and 30°-60°-90° triangle relationships, suitable for high school students (Grades 9-12).

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