https://file.aiquickdraw.com/mathbot/file/1727277186519v3s5cumc.jpg

In the given triangle \( ABC \), we see that \( P \) is the centroid of the triangle, as the medians (the red lines) of the triangle converge at point \( P \). A median of a triangle is a line drawn from a vertex to the midpoint of the opposite side. This fact gives us a useful property: the centroid divides each median into two segments, where the segment connecting the centroid to the vertex is twice as long as the segment connecting the centroid to the midpoint of the opposite side.

### Key information:
- \( AZ = 33 \) (where \( Z \) is the midpoint of side \( BC \)).
- We are to find \( AP \) and \( PZ \).

### Centroid property:
Since \( P \) divides \( AZ \) into a 2:1 ratio:
\[
AP = \frac{2}{3} \times AZ
\quad \text{and} \quad
PZ = \frac{1}{3} \times AZ
\]

### Calculations:
1. \( AP = \frac{2}{3} \times 33 = 22 \)
2. \( PZ = \frac{1}{3} \times 33 = 11 \)

Thus:
- \( AP = 22 \)
- \( PZ = 11 \)

Let me know if you need more details or have any questions.

Here are five additional questions based on this concept:
1. How do you calculate the length of the other medians in the triangle if you know all the side lengths?
2. Can the centroid ever lie outside the triangle?
3. What is the significance of the centroid in terms of the triangle’s area?
4. If \( P \) divides another median, say \( BY \), what would the lengths of \( BP \) and \( PY \) be if \( BY = 27 \)?
5. How does the centroid relate to other points of concurrency in a triangle?

**Tip:** Remember that the centroid always divides each median in the ratio 2:1, regardless of the triangle's shape.

For the following image, if AZ = 33, find AP and PZ in the triangle where P is the centroid.

Learn how to find the lengths of AP and PZ in a triangle where P is the centroid. Given that AZ = 33, use the centroid theorem to divide the median AZ into two segments in a 2:1 ratio. This problem covers key concepts of triangle geometry, centroid properties, and suitable for Grades 8-10 students.

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