Math Problem Statement

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

Solution

We are asked to find the endpoints of the median from vertex BB in triangle ABCABC. From the image, we have the following details:

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

Part A: Finding the median

The median from vertex BB will connect B(4,8)B(4, 8) to the midpoint of side ACAC.

  1. Find the midpoint of side ACAC:

The midpoint formula for two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

For A(2,2)A(2, 2) and C(8,2)C(8, 2), the midpoint is: (2+82,2+22)=(5,2)\left( \frac{2 + 8}{2}, \frac{2 + 2}{2} \right) = (5, 2)

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

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

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Math Problem Analysis

Mathematical Concepts

Geometry
Centroid of a Triangle
Midpoint Formula

Formulas

Midpoint Formula: ((x1 + x2) / 2, (y1 + y2) / 2)

Theorems

Centroid Theorem: The centroid of a triangle divides each median in a 2:1 ratio

Suitable Grade Level

Grades 9-10