Question

To prove that all three medians of a triangle meet at the same point, the medians of each side A(4,5) B(1,2) C(6,2)must be found. What are the ordered pairs of the three median bisectors? (1 point)
Responses:
A. c = (1, 2) b = (4, 5) and a = (6, 2)
B. c = (3, 4) b = (4.5, 4) and a = (4, 2)
C. c = (1.5, 1.5) b = (1, 1.5) and a = (2.5, 0)
D. c = (2.5, 3.5) b = (5, 3.5) and a = (3.5, 2)

Answer 1

To find the coordinates of the medians of triangle ABC with vertices A(4,5), B(1,2), and C(6,2), we first need the midpoints of each side.

1. **Finding the midpoints:**
- Midpoint of BC:
- B(1, 2) and C(6, 2)
- Midpoint M1 = \(\left(\frac{1+6}{2}, \frac{2+2}{2}\right) = \left(\frac{7}{2}, 2\right) = (3.5, 2)\)

- Midpoint of AC:
- A(4, 5) and C(6, 2)
- Midpoint M2 = \(\left(\frac{4+6}{2}, \frac{5+2}{2}\right) = \left(\frac{10}{2}, \frac{7}{2}\right) = (5, 3.5)\)

- Midpoint of AB:
- A(4, 5) and B(1, 2)
- Midpoint M3 = \(\left(\frac{4+1}{2}, \frac{5+2}{2}\right) = \left(\frac{5}{2}, \frac{7}{2}\right) = (2.5, 3.5)\)

2. **Finding the equations of the medians:**
- Median from A to M1:
- Points A(4, 5) and M1(3.5, 2)
- Median from B to M2:
- Points B(1, 2) and M2(5, 3.5)
- Median from C to M3:
- Points C(6, 2) and M3(2.5, 3.5)

3. **Finding the coordinates of the intersection of these medians (i.e., the centroid):**
- The centroid (G) of a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3) is given by:
\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
- Substituting our points:
\[
G = \left( \frac{4 + 1 + 6}{3}, \frac{5 + 2 + 2}{3} \right) = \left( \frac{11}{3}, \frac{9}{3} \right) = \left( \frac{11}{3}, 3 \right)
\]

Next, we match the calculated midpoints against the options provided.

For the responses:

- A: c = (1, 2), b = (4, 5), a = (6, 2)
- B: c = (3, 4), b = (4.5, 4), a = (4, 2)
- C: c = (1.5, 1.5), b = (1, 1.5), a = (2.5, 0)
- D: c = (2.5, 3.5), b = (5, 3.5), a = (3.5, 2)

None of the given choices appear to match the midpoints we calculated for the medians directly:

- The midpoints M1, M2, and M3 for the three sides should correspond to the midpoints needed for forming the medians, but they don't match any response.

The correct midpoints of the sides are:
1. \(M_1: (3.5, 2)\)
2. \(M_2: (5, 3.5)\)
3. \(M_3: (2.5, 3.5)\)

Looking at the options, it seems that option D does contain some valid midpoints. In this case, given the closest values, option D can be interpreted as the correct response for the midpoints related to mailbut might not be naturally straightforward in the comparisons.

The answer would be:
**D. c = (2.5, 3.5), b = (5, 3.5), and a = (3.5, 2)** (the closest option based on calculations and formats assumed and described).

However, if we want the actual calculated midpoints, they do not match any provided directly.