To find the coordinates of point [tex]\( N \)[/tex], we'll follow a series of steps. We start with finding the coordinates of the midpoint [tex]\( M \)[/tex] of segment [tex]\( KL \)[/tex], and then use [tex]\( M \)[/tex] to find the midpoint [tex]\( N \)[/tex] of segment [tex]\( ML \)[/tex].
1. Determine the coordinates of the midpoint [tex]\( M \)[/tex] of segment [tex]\( KL \)[/tex].
Given:
[tex]\[
K = (-7, -6) \quad \text{and} \quad L = (1, 10)
\][/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:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Plugging in the coordinates of [tex]\( K \)[/tex] and [tex]\( L \)[/tex]:
[tex]\[
M_x = \frac{-7 + 1}{2} = \frac{-6}{2} = -3
\][/tex]
[tex]\[
M_y = \frac{-6 + 10}{2} = \frac{4}{2} = 2
\][/tex]
Thus, the coordinates of point [tex]\( M \)[/tex] are:
[tex]\[
M = (-3, 2)
\][/tex]
2. Determine the coordinates of the midpoint [tex]\( N \)[/tex] of segment [tex]\( ML \)[/tex].
Given:
[tex]\[
M = (-3, 2) \quad \text{and} \quad L = (1, 10)
\][/tex]
Using the same midpoint formula:
[tex]\[
N = \left( \frac{M_x + L_x}{2}, \frac{M_y + L_y}{2} \right)
\][/tex]
Plugging in the coordinates of [tex]\( M \)[/tex] and [tex]\( L \)[/tex]:
[tex]\[
N_x = \frac{-3 + 1}{2} = \frac{-2}{2} = -1
\][/tex]
[tex]\[
N_y = \frac{2 + 10}{2} = \frac{12}{2} = 6
\][/tex]
Thus, the coordinates of point [tex]\( N \)[/tex] are:
[tex]\[
N = (-1, 6)
\][/tex]
Therefore, the coordinates of point [tex]\( N \)[/tex] are [tex]\((-1, 6)\)[/tex], which corresponds to answer choice [tex]\( \text{B} \)[/tex].
