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We are asked to find the endpoints of the median from vertex \( B \) in triangle \( ABC \). From the image, we have the following details:

- The centroid of \( \triangle ABC \) is at \( (4, 5) \).
- The vertices of \( \triangle ABC \) are:
  - \( A(2, 2) \)
  - \( B(4, 8) \)
  - \( C(8, 2) \)

### Part A: Finding the median
The median from vertex \( B \) will connect \( B(4, 8) \) to the midpoint of side \( AC \).

1. **Find the midpoint of side \( AC \):**

The midpoint formula for two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

For \( A(2, 2) \) and \( C(8, 2) \), the midpoint is:
\[
\left( \frac{2 + 8}{2}, \frac{2 + 2}{2} \right) = (5, 2)
\]

So, the median starts at \( B(4, 8) \) and ends at the midpoint \( (5, 2) \).

Thus, the median begins at \( B(4, 8) \) and ends at \( (5, 2) \).

Would you like further details on the steps or concepts? Here are five related questions:

1. How is the centroid of a triangle calculated?
2. Why does the median always pass through the centroid of a triangle?
3. How can you find the length of the median using the distance formula?
4. What is the significance of the centroid dividing the medians into a 2:1 ratio?
5. How can you find the equation of the line representing the median?

**Tip:** Remember that the centroid divides each median in a triangle into a 2:1 ratio, with the longer part being closer to the vertex.

Find the coordinates of the endpoints of the median from vertex B in triangle ABC given the centroid and the coordinates of the vertices.

Learn how to find the endpoints of the median from vertex B in triangle ABC. Given the centroid at (4, 5) and vertices at A(2, 2), B(4, 8), and C(8, 2), this problem demonstrates the use of the midpoint formula and centroid theorem. Suitable for Grades 9-10.

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