To determine the coordinates of point [tex]\( N \)[/tex], we need to follow the mathematical steps provided.
1. Firstly, we find the midpoint [tex]\( M \)[/tex] of the segment [tex]\( KL \)[/tex]. The formula for the midpoint [tex]\( M \)[/tex] of a 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 point [tex]\( K \)[/tex] as [tex]\((-7, -6)\)[/tex] and point [tex]\( L \)[/tex] as [tex]\((1, 10)\)[/tex]:
- For the x-coordinate of [tex]\( M \)[/tex]:
[tex]\[
\frac{-7 + 1}{2} = \frac{-6}{2} = -3
\][/tex]
- For the y-coordinate of [tex]\( M \)[/tex]:
[tex]\[
\frac{-6 + 10}{2} = \frac{4}{2} = 2
\][/tex]
Thus, point [tex]\( M \)[/tex] has coordinates [tex]\((-3, 2)\)[/tex].
2. Next, we find the midpoint [tex]\( N \)[/tex] of the segment [tex]\( ML \)[/tex]. Using the same midpoint formula, where [tex]\( M \)[/tex] has coordinates [tex]\((-3, 2)\)[/tex] and [tex]\( L \)[/tex] has coordinates [tex]\((1, 10)\)[/tex]:
- For the x-coordinate of [tex]\( N \)[/tex]:
[tex]\[
\frac{-3 + 1}{2} = \frac{-2}{2} = -1
\][/tex]
- For the y-coordinate of [tex]\( N \)[/tex]:
[tex]\[
\frac{2 + 10}{2} = \frac{12}{2} = 6
\][/tex]
Therefore, the coordinates of point [tex]\( N \)[/tex] are [tex]\((-1, 6)\)[/tex].
Based on our calculations, the correct option matching the coordinates of point [tex]\( N \)[/tex] is:
A. [tex]\((-1, 6)\)[/tex]
