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To solve for \( x \), \( y \), and \( z \), we can analyze the right triangles in this figure and use trigonometric ratios associated with the 30-60-90 and 45-45-90 triangles.

### Step-by-Step Solution

1. **Identify the 45-45-90 Triangle for \( x \) and \( y \):**

   - In a 45-45-90 triangle, the legs are equal, and the hypotenuse is the leg length multiplied by \( \sqrt{2} \).
   - Here, \( x \) and \( y \) are the legs, so \( x = y \).
   - The hypotenuse of this triangle is labeled as \( 14\sqrt{2} \), which gives:
     \[
     x \sqrt{2} = 14\sqrt{2}
     \]
   - Solving for \( x \):
     \[
     x = \frac{14\sqrt{2}}{\sqrt{2}} = 14
     \]
   - Therefore, \( x = 14 \) and \( y = 14 \).

2. **Identify the 30-60-90 Triangle for \( z \):**

   - In a 30-60-90 triangle, the side opposite the 30° angle (shorter leg) is half the hypotenuse, and the side opposite the 60° angle (longer leg) is the shorter leg multiplied by \( \sqrt{3} \).
   - Here, the side \( y \) (which we found to be 14) is opposite the 60° angle, so \( z \) (the side opposite the 30° angle) will be:
     \[
     z = \frac{y}{\sqrt{3}} = \frac{14}{\sqrt{3}} = \frac{14\sqrt{3}}{3}
     \]

### Final Answer
- \( x = 14 \)
- \( y = 14 \)
- \( z = \frac{14\sqrt{3}}{3} \)

Would you like further clarification on any steps?

### Additional Questions to Explore
1. How would the values change if the hypotenuse of the 45-45-90 triangle was a different length?
2. What if the 30° angle was in a different triangle configuration?
3. Can you apply these ratios to other triangles without 45-45-90 or 30-60-90 angles?
4. How would we solve if \( z \) was given instead of \( 14\sqrt{2} \)?
5. What are the other possible ways to solve this if the figure was altered?

**Tip:** Remember that knowing the properties of 45-45-90 and 30-60-90 triangles is very useful for quickly solving problems involving right triangles.

Find the value of x, y, and z in the given triangle diagram with a hypotenuse labeled 14√2 and angle measures of 45° and 30°. Answers should be in simplest radical form.

Learn how to find the values of x, y, and z in a composite right triangle diagram involving 45-45-90 and 30-60-90 triangles. This problem uses trigonometric properties and special triangle theorems to determine the side lengths in simplest radical form. Suitable for Grades 9-10 students.

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