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In this exercise, you are asked to write three different equations for \( PE \), involving the segments \( AE \), \( AP \), and others.

Looking at the triangle in the diagram:

1. \( P \) appears to be a centroid of the triangle. This means that:
   - The centroid divides each median in a 2:1 ratio, with the longer part closer to the vertex.

So the following equations for \( PE \) can be written:

### First equation:
Since the centroid divides the median into a 2:1 ratio, you can express:
\[ PE = \frac{1}{3} \times AE \]
This is provided in the first box.

### Second equation:
From the same reasoning about the centroid's position on the median:
\[ PE = \frac{1}{2} \times AP \]
This equation can be entered in the second box.

### Third equation:
The total length of the median \( AE \) can be split as the sum of two parts, \( PE \) and \( AP \), where:
\[ PE = AE - AP \]
This can go into the third box.

Would you like further explanation or details on centroids?

Here are 5 questions to deepen your understanding:
1. What is the relationship between a centroid and the medians of a triangle?
2. How does a centroid divide each median of a triangle?
3. Why does \( PE \) equal \( \frac{1}{3} \times AE \)?
4. What is the significance of the \( 2:1 \) ratio at the centroid?
5. Can the same approach be applied to all triangles?

**Tip**: Remember that the centroid divides each median into two segments where the larger segment is twice the length of the smaller one!

Use the numbers and symbols to write three different equations for PE in a triangle diagram where P is the centroid.

Explore how to write different equations for PE using the centroid properties of a triangle. This problem involves understanding medians and the centroid theorem, making it suitable for Grades 9-10 geometry students.

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