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

Question

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=(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=(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

Answer 1

To find the coordinates of the centroid (the point where all three medians of a triangle meet), we first need the coordinates of the triangle's vertices. Let's denote the vertices of the triangle as A, B, and C with coordinates A(x_A, y_A), B(x_B, y_B), and C(x_C, y_C).

The medians of a triangle are line segments from each vertex to the midpoint of the opposite side. The midpoints (M_A, M_B, M_C) of the sides BC, AC, and AB are calculated as follows:

1. Midpoint M_B of side AC: \[ M_B = \left(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}\right) \]

2. Midpoint M_C of side AB: \[ M_C = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) \]

3. Midpoint M_A of side BC: \[ M_A = \left(\frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}\right) \]

Next, we calculate the equations of the medians.

Here is the process for finding the centroid (the intersection of the medians): The centroid G of triangle ABC is given by the average of the vertices' coordinates: \[ G = \left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right) \]

Now, you have provided the responses which are ordered pairs. To determine which set of coordinates corresponds to the medians' intersections, we would need the specific coordinates of the triangle's vertices (A, B, and C).

As you have not provided those specific coordinates, it is impossible to definitively choose one of the responses. However, if you have the coordinates of A, B, and C, you can use the above process to find the midpoints and the centroid.

If you input the vertices of the triangle, we can analyze the values and select the correct median bisectors accordingly.