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The image you uploaded displays two right triangles with angles of 30°, 45°, and 60°. We are tasked with finding the exact values of side lengths \( h \) and \( b \), expressed in simplified radical form.

### For the 45°-45°-90° triangle:
- In a 45°-45°-90° triangle, the side ratio is \( 1 : 1 : \sqrt{2} \).
- Given that one leg is 5 (as seen in the diagram), the hypotenuse \( h \) can be found using the ratio:
  \[
  h = 5\sqrt{2}
  \]

### For the 30°-60°-90° triangle:
- In a 30°-60°-90° triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \), where the shortest side (opposite the 30° angle) is 3.
- To find the other side \( b \) (opposite the 60° angle), use the ratio:
  \[
  b = 3\sqrt{3}
  \]

Thus, the values are:
- \( h = 5\sqrt{2} \)
- \( b = 3\sqrt{3} \)

Would you like more details or have any questions?

Here are 5 related questions:
1. How are the side ratios determined in 45°-45°-90° triangles?
2. What is the general relationship between angles and sides in a 30°-60°-90° triangle?
3. How do you simplify radical expressions like \( 5\sqrt{2} \)?
4. Can these methods be applied to other special triangles, like equilateral triangles?
5. How do you derive the trigonometric ratios for non-special angles?

**Tip**: Memorizing the side ratios of special triangles (45°-45°-90° and 30°-60°-90°) will help you quickly solve similar problems in trigonometry.

For the right triangles shown, find the exact values of the side lengths h and b, using simplified radical form.

Learn how to find the exact side lengths in special right triangles with 30°, 45°, and 60° angles using trigonometric ratios. This problem involves simplifying radical expressions and applying known side ratios. Suitable for students in Grades 9-10.

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