Question

Use the image to answer the question.

A coordinate plane's axes range from 0 to 8, both by 1-unit increments. A triangle made up of vertices upper A(4,5), upper B (1,2), and upper C (6,2) is plotted. The coordinates of the vertices are given.

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)
Responses

c=(1.5,1.5)
, b=(1,1.5)
, and a=(2.5,0)
c equals left parenthesis 1.5 comma 1.5 right parenthesis , b equals left parenthesis 1 comma 1.5 right parenthesis , and a equals left parenthesis 2.5 comma 0 right parenthesis

c=(3,4)
, b=(4.5,4)
, and a=(4,2)
c equals left parenthesis 3 comma 4 right parenthesis , b equals left parenthesis 4.5 comma 4 right parenthesis , and a equals left parenthesis 4 comma 2 right parenthesis

c=(1,2)
, b=(4,5)
, and a=(6,2)
c equals left parenthesis 1 comma 2 right parenthesis , b equals left parenthesis 4 comma 5 right parenthesis , and a equals left parenthesis 6 comma 2 right parenthesis

c=(2.5,3.5)
, b=(5,3.5)
, and a=(3.5,2)

Answer 1

To find the ordered pairs of the three median bisectors of the triangle with vertices A(4,5), B(1,2), and C(6,2), we first need to determine the midpoints of each side of the triangle.

1. **Find the midpoints of each side:**

- Midpoint of side BC:
\[
\text{Midpoint} = \left( \frac{B_x + C_x}{2}, \frac{B_y + C_y}{2} \right) = \left( \frac{1 + 6}{2}, \frac{2 + 2}{2} \right) = \left( \frac{7}{2}, 2 \right) = (3.5, 2)
\]

- Midpoint of side AC:
\[
\text{Midpoint} = \left( \frac{A_x + C_x}{2}, \frac{A_y + C_y}{2} \right) = \left( \frac{4 + 6}{2}, \frac{5 + 2}{2} \right) = \left( 5, 3.5 \right)
\]

- Midpoint of side AB:
\[
\text{Midpoint} = \left( \frac{A_x + B_x}{2}, \frac{A_y + B_y}{2} \right) = \left( \frac{4 + 1}{2}, \frac{5 + 2}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = (2.5, 3.5)
\]

2. **Identify the medians:**
- The median from A connects A to the midpoint of BC, which is (3.5, 2).
- The median from B connects B to the midpoint of AC, which is (5, 3.5).
- The median from C connects C to the midpoint of AB, which is (2.5, 3.5).

So, the ordered pairs for the medians are:

- **Median from A**: (3.5, 2)
- **Median from B**: (5, 3.5)
- **Median from C**: (2.5, 3.5)

Checking the responses now, the correct answer based on the above calculation is:

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

This matches with none of the provided responses.

Please double-check your options, as it seems that the response with values does not correspond to the calculations above correctly.