To determine the coordinates of point [tex]\( N \)[/tex], the midpoint of segment [tex]\( ML \)[/tex], we will proceed through a series of steps.
First, we need to find the coordinates of point [tex]\( M \)[/tex], the midpoint of segment [tex]\( KL \)[/tex]. The coordinates of [tex]\( K \)[/tex] and [tex]\( L \)[/tex] are given:
- [tex]\( K(-7, -6) \)[/tex]
- [tex]\( L(1, 10) \)[/tex]
The formula for the midpoint [tex]\( M \)[/tex] of a segment connecting points [tex]\( K(x_1, y_1) \)[/tex] and [tex]\( L(x_2, y_2) \)[/tex] is:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substituting the coordinates of [tex]\( K \)[/tex] and [tex]\( L \)[/tex]:
[tex]\[
M = \left( \frac{-7 + 1}{2}, \frac{-6 + 10}{2} \right) = \left( \frac{-6}{2}, \frac{4}{2} \right) = (-3, 2)
\][/tex]
So, point [tex]\( M \)[/tex] is at [tex]\((-3, 2)\)[/tex].
Next, we find point [tex]\( N \)[/tex], the midpoint of segment [tex]\( ML \)[/tex]. Since we now have the coordinates of both [tex]\( M \)[/tex] and [tex]\( L \)[/tex]:
- [tex]\( M(-3, 2) \)[/tex]
- [tex]\( K(-7, -6) \)[/tex]
Again, using the midpoint formula:
[tex]\[
N = \left( \frac{-3 + (-7)}{2}, \frac{2 + (-6)}{2} \right) = \left( \frac{-10}{2}, \frac{-4}{2} \right) = (-5, -2)
\][/tex]
Thus, the coordinates of point [tex]\( N \)[/tex] are [tex]\((-5, -2)\)[/tex].
Based on the calculations, the correct answer is:
C. [tex]\((-5, -2)\)[/tex]
