Question

there is a triangle with 3 points, A(4,5), B(1,2), C(6,2)
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?

Answer 1

To find the medians of the triangle formed by points \( A(4, 5) \), \( B(1, 2) \), and \( C(6, 2) \), we first need to find the midpoints of each side of the triangle. The medians connect each vertex with the midpoint of the opposite side.

### Step 1: Find the Midpoints

1. **Midpoint of side BC** (between points B and C):
\[
\text{Midpoint of } BC = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) = \left( \frac{1 + 6}{2}, \frac{2 + 2}{2} \right) = \left( \frac{7}{2}, 2 \right) = \left( 3.5, 2 \right)
\]

2. **Midpoint of side AC** (between points A and C):
\[
\text{Midpoint of } AC = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{4 + 6}{2}, \frac{5 + 2}{2} \right) = \left( \frac{10}{2}, \frac{7}{2} \right) = \left( 5, 3.5 \right)
\]

3. **Midpoint of side AB** (between points A and B):
\[
\text{Midpoint of } AB = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{4 + 1}{2}, \frac{5 + 2}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = \left( 2.5, 3.5 \right)
\]

### Step 2: Find the Equations of the Medians

Next, we can calculate the equations of the medians based on the midpoints found:

1. **Median from A to midpoint of BC (3.5, 2)**:
- Points: A(4, 5) and Midpoint of BC(3.5, 2)
- Slope of median:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 5}{3.5 - 4} = \frac{-3}{-0.5} = 6
\]
- Using point-slope form:
\[
y - 5 = 6(x - 4) \implies y = 6x - 24 + 5 \implies y = 6x - 19
\]

2. **Median from B to midpoint of AC (5, 3.5)**:
- Points: B(1, 2) and Midpoint of AC(5, 3.5)
- Slope of median:
\[
m = \frac{3.5 - 2}{5 - 1} = \frac{1.5}{4} = \frac{3}{8}
\]
- Using point-slope form:
\[
y - 2 = \frac{3}{8}(x - 1) \implies y = \frac{3}{8}x + 2 - \frac{3}{8} \implies y = \frac{3}{8}x + \frac{13}{8}
\]

3. **Median from C to midpoint of AB (2.5, 3.5)**:
- Points: C(6, 2) and Midpoint of AB(2.5, 3.5)
- Slope of median:
\[
m = \frac{3.5 - 2}{2.5 - 6} = \frac{1.5}{-3.5} = -\frac{3}{7}
\]
- Using point-slope form:
\[
y - 2 = -\frac{3}{7}(x - 6) \implies y = -\frac{3}{7}x + \frac{18}{7} + 2 \implies y = -\frac{3}{7}x + \frac{32}{7}
\]

### Conclusion

The orders pairs of the midpoints (which serve as the median endpoints) are:

1. Median from \( A \) to \( \left(3.5, 2\right) \)
2. Median from \( B \) to \( \left(5, 3.5\right) \)
3. Median from \( C \) to \( \left(2.5, 3.5\right) \)

These medians will intersect at the centroid of the triangle, which can also be found as follows:

### Centroid Calculation

The centroid \( G \) of the triangle is given by the average of the vertices coordinates:
\[
G\left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right) = \left(\frac{4 + 1 + 6}{3}, \frac{5 + 2 + 2}{3}\right) = \left(\frac{11}{3}, \frac{9}{3}\right) = \left(\frac{11}{3}, 3\right)
\]

Thus, all three medians intersect at the point \( \left(\frac{11}{3}, 3\right) \).