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The given problem shows that triangle \( \triangle ABC \) is rotated around point \( P \) to create triangle \( \triangle A'B'C' \). The question asks about the relationship between the lengths of segments \( AP \) and \( A'P \).

In a rotation, corresponding points in the original and rotated triangles are equidistant from the center of rotation. Since \( P \) is the center of rotation:

\[
AP = A'P
\]

This means that the distance from point \( A \) to point \( P \) is the same as the distance from point \( A' \) to point \( P \).

Would you like further details on the properties of rotations, or do you have any other specific questions?

### Related Questions:
1. What is the relationship between \( BP \) and \( B'P \)?
2. What is the angle of rotation between triangles \( \triangle ABC \) and \( \triangle A'B'C' \)?
3. How are the angles of \( \triangle ABC \) and \( \triangle A'B'C' \) related after the rotation?
4. What happens to the lengths of sides in \( \triangle ABC \) after the rotation?
5. Can a rotation change the area of the triangle?

**Tip:** In a rotation, not only are distances from the center of rotation preserved, but also the angles between corresponding points are maintained.

What is the relationship between AP and A'P after rotating triangle ABC around point P to create triangle A'B'C'?

Explore the relationship between AP and A'P after triangle ABC is rotated around point P. This geometry problem involves key concepts like rotation and the preservation of distance. Ideal for students in Grades 9-10.

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