To find the midpoint [tex]\( C = (M_x, M_y) \)[/tex] of a line segment [tex]\(\overline{AB}\)[/tex] where the endpoints are [tex]\( A(-2, 3) \)[/tex] and [tex]\( B(1, 8) \)[/tex], we use the midpoint formula. The formula for finding the midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Given the coordinates of [tex]\(A\)[/tex] are [tex]\((A_x, A_y) = (-2, 3)\)[/tex] and of [tex]\(B\)[/tex] are [tex]\((B_x, B_y) = (1, 8)\)[/tex], we substitute these values into the midpoint formula:
For the x-coordinate of the midpoint [tex]\(M_x\)[/tex]:
[tex]\[
M_x = \frac{A_x + B_x}{2} = \frac{-2 + 1}{2} = \frac{-1}{2} = -0.5
\][/tex]
For the y-coordinate of the midpoint [tex]\(M_y\)[/tex]:
[tex]\[
M_y = \frac{A_y + B_y}{2} = \frac{3 + 8}{2} = \frac{11}{2} = 5.5
\][/tex]
Therefore, the coordinates of the midpoint [tex]\(C\)[/tex] are:
[tex]\[
M = (-0.5, 5.5)
\][/tex]
Reviewing the given choices:
1. [tex]\(M = \left( \frac{-2 + 3}{2}, \frac{1 + 8}{2} \right)\)[/tex]
2. [tex]\(M = \left( \frac{-2 + 1}{2}, \frac{3 + 8}{2} \right)\)[/tex]
3. [tex]\(M = \left( \frac{-2 - 3}{2}, \frac{1 - 8}{2} \right)\)[/tex]
4. [tex]\(M = \left( \frac{-2 - 1}{2}, \frac{3 - 8}{2} \right)\)[/tex]
The correct formula and corresponding choice is:
[tex]\[ M = \left( \frac{-2 + 1}{2}, \frac{3 + 8}{2} \right) \][/tex]
Thus, the correct answer is:
[tex]\[
M = \left( \frac{-2 + 1}{2}, \frac{3 + 8}{2} \right)
\][/tex]
