To copy a segment $PQ$ onto a line with an endpoint at $R$, follow these geometric steps:

Using a Compass and Straightedge:

1. Draw the Line with Endpoint $R$: Extend the line starting from $R$ in the desired direction.

2. Measure the Length of $PQ$:

   • Place the sharp end of the compass at point $P$.
   • Adjust the compass so that the pencil end touches point $Q$. This sets the compass to the length of $PQ$.

3. Transfer the Length to the Line Starting at $R$:

   • Without altering the compass width, place the sharp end of the compass at point $R$.
   • Draw an arc that intersects the line starting at $R$.

4. Mark the Intersection:

   • Label the intersection point of the arc and the line as $S$. This makes $RS$ equal in length to $PQ$.

5. Verify the Construction: Use a ruler to confirm that $RS = PQ$.

Result:

You have successfully copied $PQ$ onto the line starting at $R$ with $RS = PQ$.

Would you like further clarification on the geometric principles, or a diagram to illustrate the steps? Here are 5 related questions and one tip:

Questions:

1. How do you determine the accuracy of a compass-based measurement in geometry?
2. What are common errors when copying a segment using this method?
3. Can this method be adapted to copy an arc or angle instead of a line segment?
4. How would you perform this task analytically using coordinate geometry?
5. What applications does segment-copying have in real-world contexts?

Tip:

Always ensure the compass is tightened securely before transferring lengths to avoid inaccuracies during construction.