Question

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?

a. c = (3,4), b = (4.5, 4), and a = (4,2)
b. c = (1.5, 1.5), b = (1, 1.5), and a = (2.5,0)
c. c = (1,2), b = (4,5), and a = (6,2)
d. c = (2.5, 3.5), b = (5, 3.5), and a = (3.5,2)

Answer 1

To find the medians of triangle ABC with vertices A(4, 5), B(1, 2), and C(6, 2), we need to determine the points at which each median intersects the side opposite the respective vertex.

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

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

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

Now that we have the midpoints of all sides:
- Midpoint of BC at (3.5, 2)
- Midpoint of AC at (5, 3.5)
- Midpoint of AB at (2.5, 3.5)

The ordered pairs of the three median bisectors are:
- Median from A goes to (3.5, 2)
- Median from B goes to (5, 3.5)
- Median from C goes to (2.5, 3.5)

Hence, the ordered pairs of the three median bisectors correspond to:

**d. \( c = (2.5, 3.5), b = (5, 3.5), a = (3.5, 2) \)**

Note: There appears to be a mix-up in your original options with the letters tied to the ordered pairs. However, based on correct calculations, option d reflects the correct midpoints of the sides of the triangle which the medians are drawn toward.