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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=(3,4)
𝑐
=
(
3
,
4
)
, b=(4.5,4)
𝑏
=
(
4.5
,
4
)
, and a=(4,2)
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)
𝑐
=
(
2.5
,
3.5
)
, b=(5,3.5)
𝑏
=
(
5
,
3.5
)
, and a=(3.5,2)
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=(1.5,1.5)
𝑐
=
(
1.5
,
1.5
)
, b=(1,1.5)
𝑏
=
(
1
,
1.5
)
, and a=(2.5,0)
𝑎
=
(
2.5
,
0
)
c equals left parenthesis 1.5 comma 1.5 right parenthesis , b equals left parenthesis 1 comma 1.5 right p

1 answer

To find the ordered pairs of the medians of a triangle, we start with the vertices of the triangle. Let's denote the vertices of the triangle as \( A \), \( B \), and \( C \). The median is a line segment that connects a vertex to the midpoint of the opposite side.

Steps to find the medians:

  1. Find the midpoints of each side:

    • The midpoint \( M_{BC} \) of side \( BC \) can be calculated as: \[ M_{BC} = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) \]
    • The midpoint \( M_{AC} \) of side \( AC \) is: \[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) \]
    • The midpoint \( M_{AB} \) of side \( AB \) is: \[ M_{AB} = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \]

  2. Find the equations of the medians:

    • The median from vertex \( A \) to midpoint \( M_{BC} \) can be represented.
    • The median from vertex \( B \) to midpoint \( M_{AC} \) and so forth.

  3. Determine if they all meet at a common point (centroid):

    • The centroid \( G \) of the triangle, which is the point where all three medians intersect, can be computed using the formula: \[ G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right) \]

Given options and selecting the points:

Unfortunately, the ordered pairs of the medians were not clearly shared initially. Assuming we start with specific vertices of a triangle, we need to calculate the medians correctly from given vertices.

However, based on your options, it seems like you're providing specific median points based on different triangles. To confirm, each option lists three points as the midpoints and intersections of the medians.

If you can kindly provide the vertices of the triangle or specific context, I could help you compute the medians correctly as ordered pairs, and I can assist in verifying which set of points corresponds to the triangle request you have.