https://file.aiquickdraw.com/mathbot/file/1727867233559d94i7vs0.jpg

Let's work through both triangles in the image to solve for \( c \) and \( d \) using trigonometric relationships.

### Triangle 1 (left):
- Given:
  - Hypotenuse = 7
  - Angle = 30°
  - Side \( c \) is opposite the 30° angle.

In a 30°-60°-90° triangle, the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is \( \sqrt{3} \) times the shorter leg.

- Formula: 
  - \( c = \frac{1}{2} \times 7 = 3.5 \).
  - The other leg is \( 3.5 \times \sqrt{3} = 3.5\sqrt{3} \).

So, the exact value for \( c \) is:

\[
c = 3.5
\]

### Triangle 2 (right):
- Given:
  - Hypotenuse = 6
  - Angle = 45°
  - Side \( d \) is adjacent to the 45° angle.

In a 45°-45°-90° triangle, both legs are equal, and the hypotenuse is \( \sqrt{2} \) times one leg.

- Formula:
  - \( d = \frac{6}{\sqrt{2}} = 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2} \).

So, the exact value for \( d \) is:

\[
d = 3\sqrt{2}
\]

### Final Answers:
- \( c = 3.5 \)
- \( d = 3\sqrt{2} \)

Would you like more details on any of the steps or have any questions?

Here are 5 related questions you could explore:
1. How do you calculate the sides of a 30°-60°-90° triangle?
2. Why is the hypotenuse in a 45°-45°-90° triangle \( \sqrt{2} \) times the leg?
3. How would the calculations change if the hypotenuse was different in either triangle?
4. What is the significance of using simplified radical form in trigonometry?
5. Can these triangles be used to solve real-world problems, such as calculating distances or heights?

**Tip:** Remember that special right triangles (like 30°-60°-90° and 45°-45°-90°) have fixed side ratios, which makes solving for sides straightforward.

For the right triangles below, find the exact values of the side lengths c and d.

Find the exact values of the side lengths c and d in a 30°-60°-90° and 45°-45°-90° right triangle using trigonometric properties. Learn how to simplify radical forms and apply special triangle theorems.

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