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To prove that all three medians of a triangle meet at the same point, the medians of each side must be found. What are the ordered pairs of the three median bisectors?

(1 point) A(4,5) B(1,2) C(6,2)

1 answer

To find the medians of a triangle, we first need to find the midpoints of each of the sides of the triangle defined by the points \( A(4,5) \), \( B(1,2) \), and \( C(6,2) \). The median of a triangle is a line segment from a vertex to the midpoint of the opposite side.

  1. Finding the midpoints of each side:

    • Midpoint of side BC: \[ \text{Midpoint} = \left(\frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}\right) = \left(\frac{1 + 6}{2}, \frac{2 + 2}{2}\right) = \left(\frac{7}{2}, 2\right) = (3.5, 2) \]

    • Midpoint of side AC: \[ \text{Midpoint} = \left(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}\right) = \left(\frac{4 + 6}{2}, \frac{5 + 2}{2}\right) = \left(\frac{10}{2}, \frac{7}{2}\right) = (5, 3.5) \]

    • Midpoint of side AB: \[ \text{Midpoint} = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) = \left(\frac{4 + 1}{2}, \frac{5 + 2}{2}\right) = \left(\frac{5}{2}, \frac{7}{2}\right) = (2.5, 3.5) \]

  2. Ordered pairs of the midpoints:

    • Midpoint of BC: \( (3.5, 2) \)
    • Midpoint of AC: \( (5, 3.5) \)
    • Midpoint of AB: \( (2.5, 3.5) \)

  3. Finding the equations of the medians:

    • Median from A to midpoint of BC (3.5, 2): The slope of line \( A \) to \( \text{Midpoint of BC} \): \[ \text{slope} = \frac{2 - 5}{3.5 - 4} = \frac{-3}{-0.5} = 6 \] Equation of the median is: \[ y - 5 = 6(x - 4) \quad \Rightarrow \quad y = 6x - 24 + 5 = 6x - 19 \]

    • Median from B to midpoint of AC (5, 3.5): Slope: \[ \text{slope} = \frac{3.5 - 2}{5 - 1} = \frac{1.5}{4} = \frac{3}{8} \] Equation of this median: \[ y - 2 = \frac{3}{8}(x - 1) \quad \Rightarrow \quad y = \frac{3}{8}x + 2 - \frac{3}{8} = \frac{3}{8}x + \frac{13}{8} \]

    • Median from C to midpoint of AB (2.5, 3.5): Slope: \[ \text{slope} = \frac{3.5 - 2}{2.5 - 6} = \frac{1.5}{-3.5} = -\frac{3}{7} \] Equation of this median: \[ y - 2 = -\frac{3}{7}(x - 6) \quad \Rightarrow \quad y = -\frac{3}{7}x + \frac{18}{7} + 2 = -\frac{3}{7}x + \frac{32}{7} \]

So the original question related to finding the ordered pairs of the three median bisectors gives us:

  • Midpoint of BC: \( (3.5, 2) \)
  • Midpoint of AC: \( (5, 3.5) \)
  • Midpoint of AB: \( (2.5, 3.5) \)