https://file.aiquickdraw.com/mathbot/file/172615665766203kctck5.jpg

https://file.aiquickdraw.com/mathbot/file/1726156657664kvmupp1s.jpg

From the image you've uploaded, we are looking at a geometric question involving the rotation of a triangle.

In the second image:
- **Triangle \( \triangle ABC \)** is rotated about a point \( P \), forming a new triangle \( \triangle A'B'C' \).
- The question is asking for the relationship between the segments \( AP \) and \( A'P \), given the rotation.

### Understanding the options in the first image:
1. **A. The segments are parallel to each other.**
   - This would not be correct in the context of rotation. Rotating a figure around a point doesn’t preserve parallelism between corresponding segments.

2. **B. The segments are perpendicular to each other.**
   - This might be true only if the rotation is specifically 90°, but it’s not generally true for all rotations.

3. **C. The segments are congruent to each other.**
   - **This is correct**. Rotating a figure preserves the distances between corresponding points, meaning \( AP = A'P \). This property holds for any type of rotation, as rotation is a rigid transformation (it preserves lengths).

4. **D. The image segment is twice the pre-image segment.**
   - This would be the case in a dilation transformation, not a rotation.

### Conclusion:
The correct answer is **C. The segments are congruent to each other**, because the distance between a point and the center of rotation remains the same after a rotation.

Would you like more details or have any questions?

### Here are 5 related questions for further exploration:
1. What are the properties of a rigid transformation?
2. How do rotations affect the orientation of a shape?
3. What is the difference between rotation and reflection in geometry?
4. How does the distance between points change in dilation versus rotation?
5. What is the relationship between the angle of rotation and the orientation of segments?

### Tip:
In geometry, remember that rotations preserve distances and angles, making them "rigid transformations." This means shapes retain their size and shape after being rotated.

What is the relationship between segment AP and segment A'P after a rotation of triangle ABC around point P?

Explore how rotation affects the relationship between segments in geometry. Specifically, find out how the distance between points AP and A'P remains congruent during a rigid transformation like rotation. Suitable for Grades 9-11 students studying rotation properties.

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