From your uploaded image, the task is to copy the segment $PQ$ to a line with an endpoint $R$. Here is how you can complete this construction step by step:

Steps to Copy the Segment:

1. Set the Compass to the Length of $PQ$:

   • Place the compass point on $P$.
   • Adjust the compass width to exactly match the distance to $Q$.

2. Transfer the Length to Line $R$:

   • Without changing the compass width, move the compass to the endpoint $R$.
   • Place the compass point on $R$ and draw an arc that crosses the line.

3. Mark the Intersection Point:

   • The point where the arc intersects the line is the endpoint of the new segment.
   • Label this point as $S$ (or any other name if specified).

4. Draw the Segment $RS$:

   • Use a straightedge to draw a line segment from $R$ to the intersection point, completing the copy of $PQ$.

5. Verification:

   • Measure $RS$ to confirm that its length matches $PQ$.

Would you like a more detailed explanation or need clarification on any step?

Here are five related questions for further exploration:

1. Why is it necessary to keep the compass width unchanged during the construction?
2. How can you verify that the new segment is exactly equal to the original without a ruler?
3. What are some real-world applications of segment copying in construction and design?
4. Can you copy a segment without a compass? If so, how?
5. What role do congruence and precision play in geometric constructions?

Tip: Always ensure your compass and straightedge are properly aligned for accurate constructions!