https://file.aiquickdraw.com/mathbot/file/1730463827956xbmxe7ax.jpg

To solve for \( x \) in this diagram, we can approach the problem as a multi-step application of the Pythagorean Theorem.

### Step-by-Step Solution

1. **Identify Right Triangles:**  
   The diagram contains two right triangles. The smaller triangle has sides of 2 and 5, while the larger triangle has hypotenuse \( x \) and leg lengths of 7 and the hypotenuse of the smaller triangle (which we need to find).

2. **Calculate Hypotenuse of the Smaller Triangle:**  
   For the smaller triangle with legs 2 and 5, let \( c \) be the hypotenuse:
   \[
   c = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}
   \]

3. **Use the Larger Triangle:**  
   Now we can apply the Pythagorean Theorem to the larger triangle with sides 7 and \( \sqrt{29} \), and hypotenuse \( x \):
   \[
   x = \sqrt{7^2 + (\sqrt{29})^2} = \sqrt{49 + 29} = \sqrt{78}
   \]

4. **Final Answer in Simplest Radical Form:**  
   So, the answer is:
   \[
   x = \sqrt{78}
   \]

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

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Here are five related questions to expand your understanding:
1. How do we simplify square roots, like \( \sqrt{78} \)?
2. What would change if the dimensions of the smaller triangle were different?
3. How does the Pythagorean Theorem apply to non-right triangles?
4. What are some real-world applications of the Pythagorean Theorem?
5. How would you solve this if only one triangle’s side lengths were given?

**Tip:** When solving multi-step problems, break them down into individual triangles to apply the Pythagorean Theorem accurately.

Solve for x. Leave your answer in simplest radical form.

Learn how to solve for x in a multi-step Pythagorean Theorem problem involving two right triangles. This problem requires calculating hypotenuses and applying the Pythagorean Theorem to find the solution in simplest radical form. Suitable for students in Grades 8-10.

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