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In the diagram, you are given a triangle with segment \( EH \), and you are tasked with finding the length of \( EH \). Here's the breakdown of the problem:

- You have a right-angled triangle at point \( H \).
- \( EH = u \) is one leg of the right triangle.
- \( HG = 8u - 63 \) is the other leg of the right triangle.
- The hypotenuse \( FG \) is marked as congruent to \( EF \), indicating \( FG = EF \).
  
Since the right triangle formed at \( H \) gives us two legs and we need to find \( EH \), we can apply the Pythagorean theorem:

\[
EH^2 + HG^2 = EG^2
\]

Where:
- \( EH = u \),
- \( HG = 8u - 63 \),
- \( EG = FG = EF \) (since EF and FG are marked as equal).

To solve this, additional information about \( EG \) or \( EF \) might be required. Do you have any values or relationships given for \( EG \) or \( EF \) that can help in the calculation?

Would you like me to proceed with assumptions, or clarify further? 

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Here are 5 related questions:
1. What is the length of \( EF \) or \( FG \)?
2. Is there any specific equation involving \( u \) that wasn't shown here?
3. Are there any angle measures or other side lengths available in the original problem?
4. Can you provide more information on the context of this geometry problem?
5. Are we solving for \( u \) first, or is \( u \) already known?

**Tip**: Always make sure all sides and angles in a triangle problem are labeled clearly to simplify the use of geometric rules like the Pythagorean theorem.

What is EH? (Given the triangle with segments EH, HG, and FG where EH = u, HG = 8u - 63, and FG = EF.)

This problem involves finding the length of EH in a right triangle using the Pythagorean theorem. Given the expressions for the two legs (EH = u, HG = 8u - 63) and equal lengths of FG and EF, solve for EH with a detailed breakdown of the solution. Suitable for students in Grades 9-11.

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