To determine the coordinates of point [tex]\( C \)[/tex], the midpoint of [tex]\(\overline{AB}\)[/tex] where [tex]\( A(-2, 3) \)[/tex] and [tex]\( B(1, 8) \)[/tex], follow these steps:
1. The formula for finding the midpoint [tex]\(C\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
2. Plug in the coordinates of the points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- For [tex]\( A = (-2, 3) \)[/tex], we have [tex]\( x_1 = -2 \)[/tex] and [tex]\( y_1 = 3 \)[/tex].
- For [tex]\( B = (1, 8) \)[/tex], we have [tex]\( x_2 = 1 \)[/tex] and [tex]\( y_2 = 8 \)[/tex].
3. Calculate the [tex]\( x \)[/tex]-coordinate of the midpoint:
[tex]\[
C_x = \frac{-2 + 1}{2} = \frac{-1}{2} = -0.5
\][/tex]
4. Calculate the [tex]\( y \)[/tex]-coordinate of the midpoint:
[tex]\[
C_y = \frac{3 + 8}{2} = \frac{11}{2} = 5.5
\][/tex]
5. Combine these coordinates to find the midpoint [tex]\( C \)[/tex]:
[tex]\[
C = (-0.5, 5.5)
\][/tex]
Given the multiple-choice options:
- [tex]\(M=\left(\frac{-2+3}{2}, \frac{1+8}{2}\right)\)[/tex] gives [tex]\( (0.5, 4.5) \)[/tex]
- [tex]\(M=\left(\frac{-2+1}{2}, \frac{3+8}{2}\right)\)[/tex] gives [tex]\( (-0.5, 5.5) \)[/tex]
- [tex]\(M=\left(\frac{-2-3}{2}, \frac{1-8}{2}\right)\)[/tex] gives [tex]\( (-2.5, -3.5) \)[/tex]
- [tex]\(M=\left(\frac{-2-1}{2}, \frac{3-8}{2}\right)\)[/tex] gives [tex]\( (-1.5, -2.5) \)[/tex]
The correct option that matches the computed midpoint [tex]\( (-0.5, 5.5) \)[/tex] 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]
