To find the location of point [tex]\( B \)[/tex], let's use the midpoint formula. The midpoint formula for two points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Given that point [tex]\( M = (-1, 8) \)[/tex] is the midpoint of [tex]\( \overline{A B} \)[/tex] and point [tex]\( A = (2, 6) \)[/tex], we need to find the coordinates of point [tex]\( B = (x_2, y_2) \)[/tex].
Let's start with the [tex]\( x \)[/tex]-coordinate:
[tex]\[ -1 = \frac{2 + x_2}{2} \][/tex]
To solve for [tex]\( x_2 \)[/tex], we need to isolate [tex]\( x_2 \)[/tex]:
1. Multiply both sides by 2:
[tex]\[ -2 = 2 + x_2 \][/tex]
2. Subtract 2 from both sides:
[tex]\[ x_2 = -4 \][/tex]
Now, let's find the [tex]\( y \)[/tex]-coordinate:
[tex]\[ 8 = \frac{6 + y_2}{2} \][/tex]
To solve for [tex]\( y_2 \)[/tex], we perform similar steps:
1. Multiply both sides by 2:
[tex]\[ 16 = 6 + y_2 \][/tex]
2. Subtract 6 from both sides:
[tex]\[ y_2 = 10 \][/tex]
Therefore, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (-4, 10) \)[/tex].
So, the correct answer is:
[tex]\[ (-4,10) \][/tex]
