To find the location of point [tex]\( B \)[/tex], let's use the midpoint formula. The midpoint [tex]\( M \)[/tex] of a line segment [tex]\(\overline{AB}\)[/tex] with endpoints [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:
- Point [tex]\( A \)[/tex] is located at [tex]\((2, 6)\)[/tex]
- Point [tex]\( M \)[/tex] is located at [tex]\((-1, 8)\)[/tex]
We need to find the coordinates of point [tex]\( B \)[/tex].
Let's denote point [tex]\( B \)[/tex] as [tex]\((x_2, y_2)\)[/tex].
The formula gives us two equations, one for the x-coordinates and one for the y-coordinates:
[tex]\[
M_x = \frac{x_1 + x_2}{2}
\][/tex]
[tex]\[
M_y = \frac{y_1 + y_2}{2}
\][/tex]
Substitute the given coordinates of [tex]\(A\)[/tex] and [tex]\(M\)[/tex]:
[tex]\[
-1 = \frac{2 + x_2}{2}
\][/tex]
[tex]\[
8 = \frac{6 + y_2}{2}
\][/tex]
We solve for [tex]\(x_2\)[/tex] and [tex]\(y_2\)[/tex] from these equations.
### Solving for [tex]\(x_2\)[/tex]:
Multiply both sides of the equation by 2:
[tex]\[
2 \cdot -1 = 2 + x_2
\][/tex]
[tex]\[
-2 = 2 + x_2
\][/tex]
Subtract 2 from both sides:
[tex]\[
x_2 = -4
\][/tex]
### Solving for [tex]\(y_2\)[/tex]:
Multiply both sides of the equation by 2:
[tex]\[
2 \cdot 8 = 6 + y_2
\][/tex]
[tex]\[
16 = 6 + y_2
\][/tex]
Subtract 6 from both sides:
[tex]\[
y_2 = 10
\][/tex]
Thus, the coordinates of point [tex]\( B \)[/tex] are [tex]\((-4, 10)\)[/tex].
So, the correct location of point [tex]\( B \)[/tex] is:
[tex]\[
\boxed{(-4, 10)}
\][/tex]
