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.