https://file.aiquickdraw.com/mathbot/file/1729941232573erb9q23c.jpg

To solve this problem, let's analyze the given triangle and apply trigonometric relationships. 

### Information Provided:
- The figure shows a right triangle with sides \( w \), \( x \), \( y \), and \( z \).
- \( \angle W = 52^\circ \).
- The length of side \( WZ = 6 \).
- The hypotenuse \( WY = 61 \).

We are asked to find the lengths of \( w \), \( x \), \( y \), and \( z \) and round answers to the nearest tenth.

### Solution Steps:
Since \( \triangle WYZ \) is a right triangle:
1. **Identify sides**:
   - \( w = WZ \) is the opposite side of \( \angle W \).
   - \( x = ZY \) is the adjacent side to \( \angle W \).
   - \( y = WY \) is the hypotenuse.

### Step 1: Calculate \( x \) and \( z \)
We can use trigonometric ratios with the given information:

#### Finding \( x \) (Adjacent side):
Using the cosine function:
\[
\cos(52^\circ) = \frac{x}{61}
\]
\[
x = 61 \cdot \cos(52^\circ)
\]

#### Finding \( z \) (Opposite side):
Using the sine function:
\[
\sin(52^\circ) = \frac{z}{61}
\]
\[
z = 61 \cdot \sin(52^\circ)
\]

Let's calculate these values.

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Calculating...

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### Results:
1. **\( w \)**: Given \( w = 6 \).
2. **\( x \approx 61 \cdot \cos(52^\circ) \approx 37.6 \)**
3. **\( y = 61 \) (given)**
4. **\( z \approx 61 \cdot \sin(52^\circ) \approx 48.1 \)**

### Summary:
- \( w = 6 \)
- \( x \approx 37.6 \)
- \( y = 61 \)
- \( z \approx 48.1 \)

Would you like further details or explanations on any of these steps?

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### Additional Questions:
1. How would the values change if \( \angle W \) was different?
2. How do trigonometric ratios help in finding unknown sides of a right triangle?
3. Why is it necessary to use the hypotenuse for sine and cosine calculations?
4. What other trigonometric function could be used if the values of \( x \) and \( w \) were known?
5. How can we verify the results using the Pythagorean theorem?

### Tip:
Always check if the angle and sides are in the right triangle setup before applying trigonometric functions for calculations.

Find the lengths of w, x, y, and z shown in the figure below if wy = 61. Round your answers to the nearest tenth.

Learn how to find the lengths of sides in a right triangle using trigonometric ratios. This problem includes given angle measures and side lengths, applying cosine and sine formulas. Ideal for students in Grades 9-10.

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