Sure, let's determine the coordinates of point [tex]\( N \)[/tex] using the information provided.
Given:
1. Midpoint [tex]\( P \)[/tex] is [tex]\( (-4, 6) \)[/tex].
2. Point [tex]\( M \)[/tex] is [tex]\( (8, -2) \)[/tex].
First, recall the midpoint formula. The midpoint [tex]\( P \)[/tex] of a line segment with endpoints [tex]\( M(x_1, y_1) \)[/tex] and [tex]\( N(x_2, y_2) \)[/tex] is given by:
[tex]\[ P \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
In this case:
[tex]\[ P \left( -4, 6 \right) \][/tex]
[tex]\[ M \left( 8, -2 \right) \][/tex]
We need to find the coordinates of [tex]\( N \)[/tex], say [tex]\( (x_2, y_2) \)[/tex].
From the midpoint formula, we have two equations:
1. [tex]\[ -4 = \frac{8 + x_2}{2} \][/tex]
2. [tex]\[ 6 = \frac{-2 + y_2}{2} \][/tex]
Solve for [tex]\( x_2 \)[/tex]:
[tex]\[ -4 = \frac{8 + x_2}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ -8 = 8 + x_2 \][/tex]
Subtract 8 from both sides:
[tex]\[ -16 = x_2 \][/tex]
So, the x-coordinate of point [tex]\( N \)[/tex] is [tex]\(-16\)[/tex].
Solve for [tex]\( y_2 \)[/tex]:
[tex]\[ 6 = \frac{-2 + y_2}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 12 = -2 + y_2 \][/tex]
Add 2 to both sides:
[tex]\[ 14 = y_2 \][/tex]
So, the y-coordinate of point [tex]\( N \)[/tex] is [tex]\( 14 \)[/tex].
Therefore, the coordinates of point [tex]\( N \)[/tex] are [tex]\( (-16, 14) \)[/tex].
