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To solve for the exact lengths of sides \( b \) and \( d \) in the right triangles, we can apply trigonometric ratios and the Pythagorean theorem. Let's work through each triangle separately.

### Triangle 1:
- Given: 
  - One angle is \( 45^\circ \).
  - The adjacent side to this angle (the leg of the triangle) is \( 6 \).
  - The triangle is a \( 45^\circ-45^\circ-90^\circ \) triangle.

For a \( 45^\circ-45^\circ-90^\circ \) triangle, the sides are in the ratio \( 1:1:\sqrt{2} \).

1. **Finding \( b \):**
   - The hypotenuse \( b \) can be found using the ratio:
   \[
   b = 6 \times \sqrt{2}
   \]

### Triangle 2:
- Given:
  - One angle is \( 60^\circ \).
  - The side opposite the \( 60^\circ \) angle is \( 4 \).
  - This is a \( 30^\circ-60^\circ-90^\circ \) triangle.

For a \( 30^\circ-60^\circ-90^\circ \) triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \).

1. **Finding \( d \):**
   - The hypotenuse \( d \) can be found using the ratio:
   \[
   d = 4 \times 2 = 8
   \]

### Final Answers:
- \( b = 6\sqrt{2} \)
- \( d = 8 \)

If you have any questions or need further clarification, feel free to ask!

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Here are five related questions for further exploration:
1. How do you derive the side ratios for special right triangles?
2. What is the significance of the Pythagorean theorem in solving right triangles?
3. How would you solve for the sides if the given angles were different?
4. What other methods could be used to solve for the sides of a right triangle?
5. Can you apply these principles to non-right triangles?

**Tip:** Always double-check your angle measures and side lengths to ensure accuracy when applying trigonometric ratios!

En los siguientes triángulos rectángulos, hallar las longitudes exactas de los lados b y d.

Learn how to solve for the exact lengths of sides b and d in special right triangles, using trigonometric ratios for 45°-45°-90° and 30°-60°-90° triangles. The solution includes simplified radical forms. Suitable for students in Grades 8-10.

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