Math Problem Statement

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

Solution

In the given triangle ABCABC, we see that PP is the centroid of the triangle, as the medians (the red lines) of the triangle converge at point PP. 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=33AZ = 33 (where ZZ is the midpoint of side BCBC).
  • We are to find APAP and PZPZ.

Centroid property:

Since PP divides AZAZ into a 2:1 ratio:

\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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Centroid
Triangle Properties

Formulas

Centroid divides each median in a 2:1 ratio

Theorems

Centroid Theorem
Properties of Medians

Suitable Grade Level

Grades 8-10