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The problem provided is related to the properties of triangles, specifically dealing with the centroid and a midpoint.

### Problem Recap:
- Triangle \( \triangle ABC \) has centroid \( P \).
- \( M \) is the midpoint of side \( BC \).
- The distance \( MP = 252 \, \text{cm} \) is given.
- The goal is to find the length of \( AP \).

### Key Concept:
The centroid \( P \) of a triangle divides the median from any vertex to the opposite side in a 2:1 ratio. In other words, if \( M \) is the midpoint of \( BC \), then:
\[ \frac{AP}{MP} = 2 \]

This implies:
\[ AP = 2 \times MP \]

### Solution:
Given \( MP = 252 \, \text{cm} \), we calculate:
\[
AP = 2 \times 252 = 504 \, \text{cm}
\]

Therefore, the length of \( AP \) is **504 cm**, corresponding to the third option.

Would you like further details or have any questions? 

### 5 Related Questions:
1. What is the centroid of a triangle, and how is it found?
2. How does the centroid divide each median in a triangle?
3. Can the centroid of a triangle be located outside the triangle?
4. How do you find the length of a median in a triangle?
5. In what types of triangles is the centroid also the orthocenter?

### Tip:
In any triangle, the centroid divides each median into two parts, with the section closest to the vertex being twice as long as the section closest to the opposite side.

The centroid of ΔABC is point P. Point M is the midpoint of BC, and MP = 252 cm. What is the length of AP?

In this problem, you will calculate the length of AP in ΔABC given that the centroid divides the median in a 2:1 ratio. The distance MP is 252 cm, and you can use the key formula AP = 2 * MP to find AP.

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