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.