https://file.aiquickdraw.com/mathbot/file/1730121206888sxfsljve.jpg

To solve for \( DG \) using the Pythagorean Theorem and determine \( FG \) and \( BG \), let's go through the steps based on the information in the diagram:

1. **Analyzing Triangle \( BGD \):**  
   Since \( BG = FG \) (indicated by the marks on the segments), triangle \( BGD \) is an isosceles right triangle with \( \angle BGD = 90^\circ \). This implies that \( BG = DG \).

2. **Using the Pythagorean Theorem in Triangle \( BGC \):**  
   Given:
   - \( BC = 17 \)
   - \( GC = 15 \)
   
   We can find \( BG \) by applying the Pythagorean Theorem:
   \[
   BG = \sqrt{BC^2 - GC^2} = \sqrt{17^2 - 15^2}
   \]
   
3. **Calculate \( BG \):**
   \[
   BG = \sqrt{289 - 225} = \sqrt{64} = 8
   \]
   
4. **Conclusion:**
   - Since \( BG = DG \) in the isosceles right triangle, \( DG = 8 \).
   - Also, \( FG = BG = 8 \).

So, the answers are:
- \( DG = 8 \)
- \( FG = BG = 8 \)

Would you like a detailed explanation of any step, or do you have further questions?

### Related Questions
1. How do we apply the Pythagorean Theorem in different types of triangles?
2. Why are \( BG \) and \( FG \) equal in this diagram?
3. Can we use trigonometry to find angles in such geometric figures?
4. How would we find the length of \( EF \) or \( ED \) if they were required?
5. What other properties of isosceles triangles help in solving problems like these?

### Tip
Always check for right angles and symmetry in geometric diagrams; they often simplify calculations significantly.

Use the Pythagorean Theorem to determine DG. So FG & BG both = ?

This problem demonstrates how to use the Pythagorean Theorem to calculate side lengths in a right triangle, specifically solving for DG, FG, and BG in the given geometric diagram. The solution involves applying geometry principles and understanding properties of isosceles triangles.

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