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To construct a segment congruent to segment \(\overline{RS}\) with endpoint \(P\) using a compass and straightedge, follow these steps:

### Steps:

1. **Set the compass**: Open the compass to the length of segment \(\overline{RS}\).
   - Place the compass point on \(R\) and adjust it so that the pencil touches point \(S\). This will copy the length of \(\overline{RS}\).

2. **Move the compass to point \(P\)**: Without changing the compass width, place the compass point on \(P\).

3. **Draw an arc**: Draw an arc from point \(P\) that will mark where the other endpoint of the congruent segment should be.

4. **Mark the new endpoint**: Label the point where the arc intersects your workspace as the new endpoint of the segment congruent to \(\overline{RS}\).

5. **Draw the new segment**: Use the straightedge to draw a line from point \(P\) to the new point, creating the segment.

This segment will be congruent to \(\overline{RS}\), meaning it will have the same length.

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Would you like a detailed explanation of any step or further clarification? 

Here are 5 related questions to explore:
1. How can we verify that two segments are congruent using mathematical principles?
2. What is the importance of congruence in geometric constructions?
3. Can a similar construction be done to copy angles?
4. How can you construct a perpendicular bisector of a given segment using a compass and straightedge?
5. What is the difference between congruence and similarity in geometry?

**Tip**: Always keep the compass width unchanged while transferring lengths to maintain precision.

Use the compass and straightedge to construct a segment congruent to RS with endpoint P.

Learn how to construct a segment congruent to a given segment using a compass and straightedge. This geometric construction problem demonstrates congruence and precision. Suitable for students in Grades 6-10.

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