To find the coordinates of point [tex]\(N\)[/tex], we need to follow a series of steps carefully. Let's break it down step-by-step:
1. Determine the midpoint [tex]\(M\)[/tex] of segment [tex]\(KL\)[/tex]:
- Point [tex]\(K\)[/tex] has coordinates [tex]\((-7, -6)\)[/tex].
- Point [tex]\(L\)[/tex] has coordinates [tex]\((1, 10)\)[/tex].
The formula to find the midpoint [tex]\(M\)[/tex] between 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]
Plugging in the coordinates of points [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]
Therefore, the coordinates of point [tex]\(M\)[/tex] are [tex]\((-3, 2)\)[/tex].
2. Determine the midpoint [tex]\(N\)[/tex] of segment [tex]\(ML\)[/tex]:
- Point [tex]\(M\)[/tex] has coordinates [tex]\((-3, 2)\)[/tex].
- Point [tex]\(L\)[/tex] has coordinates [tex]\((1, 10)\)[/tex].
Again, using the midpoint formula:
[tex]\[
N = \left( \frac{M_x + L_x}{2}, \frac{M_y + L_y}{2} \right)
\][/tex]
Plugging in the coordinates of points [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]
Therefore, the coordinates of point [tex]\(N\)[/tex] are [tex]\((-1, 6)\)[/tex].
So, the coordinates of point [tex]\(N\)[/tex] are:
[tex]\[ \boxed{(-1, 6)} \][/tex]
