Question

To prove that all three medians of a triangle meet at the same point, the medians of each side must be found (A(4,5) B(1,2) C(6,2)). 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=(2.5,3.5) , b=(5,3.5) , and a=(3.5,2) c equals left parenthesis 2.5 comma 3.5 right parenthesis , b equals left parenthesis 5 comma 3.5 right parenthesis , and a equals left parenthesis 3.5 comma 2 right parenthesis c=(1,2) , b=(4,5) , and a=(6,2)

Answer 1

To find the coordinates of the midpoints of each side of triangle ABC, we start with the given vertices:

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

The midpoints of each side of the triangle can be calculated as follows:

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

2. **Midpoint of AC**:
\[
M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{4 + 6}{2}, \frac{5 + 2}{2} \right) = \left( 5, \frac{7}{2} \right) = \left( 5, 3.5 \right)
\]

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

Now we will derive the coordinates of the medians, which connect each vertex of the triangle to the opposite midpoint.

The median from vertex A(4, 5) to midpoint \(M_{BC}(3.5, 2)\):
- Not needed for midpoint calculations.

The median from vertex B(1, 2) to midpoint \(M_{AC}(5, 3.5)\):
- Not needed for midpoint calculations.

The median from vertex C(6, 2) to midpoint \(M_{AB}(2.5, 3.5)\):
- Not needed for midpoint calculations.

Now, let's summarize the midpoints used to define the medians:
1. Midpoint \(M_{BC} = (3.5, 2)\)
2. Midpoint \(M_{AC} = (5, 3.5)\)
3. Midpoint \(M_{AB} = (2.5, 3.5)\)

Thus the points corresponding to the three median bisectors are:
- a = \(M_{AB} = (2.5, 3.5)\)
- b = \(M_{AC} = (5, 3.5)\)
- c = \(M_{BC} = (3.5, 2)\)

We need to check which of the answer options corresponds to these results:

The closest option by value is:
- \(c=(3, 4), b=(4.5, 4), a=(4, 2)\)

This appears incorrect. Therefore, reiterating the values derived above, consistent midpoints are:

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

Which does not match any option. Thus, no answer correctly matches derived outcomes. Further scrutiny might be warranted for incorrectly formatted calculation or mismatched defined points. The median point must align within the triangle format under specified geometry rules.

**Final verification: The answer is none of the present options. 각 medians’ defined values provide a distorted outcome in all representations.**