To find the coordinates of point [tex]\( N \)[/tex], we'll begin by determining the midpoint [tex]\( M \)[/tex] of segment [tex]\( KL \)[/tex], and then use [tex]\( M \)[/tex] to find [tex]\( N \)[/tex].
1. Find the coordinates of the midpoint [tex]\( M \)[/tex] of segment [tex]\( KL \)[/tex]:
- The midpoint formula for two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
- Given [tex]\( K(-7, -6) \)[/tex] and [tex]\( L(1, 10) \)[/tex], apply the formula:
[tex]\[
M = \left( \frac{-7 + 1}{2}, \frac{-6 + 10}{2} \right)
\][/tex]
[tex]\[
M = \left( \frac{-6}{2}, \frac{4}{2} \right)
\][/tex]
[tex]\[
M = (-3.0, 2.0)
\][/tex]
2. Find the coordinates of the midpoint [tex]\( N \)[/tex] of segment [tex]\( ML \)[/tex]:
- Using the midpoint formula again for points [tex]\( M(-3.0, 2.0) \)[/tex] and [tex]\( L(1, 10) \)[/tex]:
[tex]\[
N = \left( \frac{-3.0 + 1}{2}, \frac{2.0 + 10}{2} \right)
\][/tex]
[tex]\[
N = \left( \frac{-2.0}{2}, \frac{12.0}{2} \right)
\][/tex]
[tex]\[
N = (-1.0, 6.0)
\][/tex]
Therefore, the coordinates of point [tex]\( N \)[/tex] are [tex]\((-1, 6)\)[/tex], which corresponds to option A.
Answer: A. [tex]\((-1, 6)\)[/tex]
