Answer:
[tex](-10,\, -7)[/tex].
Step-by-step explanation:
If [tex]A\; (x_{a},\, y_{a})[/tex] and [tex]B\; (x_{b},\, y_{b})[/tex] are the two endpoints of a line segment, the midpoint of that line segment will be at:
[tex]\begin{aligned}\left(\frac{x_{a} + x_{b}}{2},\, \frac{y_{a} + y_{b}}{2}\right)\end{aligned}[/tex].
In other words, the [tex]x[/tex]-coordinate of the midpoint will be [tex](1/2)\, (x_{a} + x_{b})[/tex] while the [tex]y[/tex]-coordinate of the midpoint will be [tex](1/2)\, (y_{a} + y_{b})[/tex].
In this question, it is given that point [tex]A[/tex] is at [tex](4,\, 1)[/tex], such that [tex]x_{a} = 4[/tex] and [tex]y_{a} = 1[/tex], while [tex]x_{b}[/tex] and [tex]y_{b}[/tex] need to be found.
The midpoint is at [tex](-3,\, -3)[/tex], with an [tex]x[/tex]-coordinate of [tex](-3)[/tex] and a [tex]y[/tex]-coordinate of [tex](-3)[/tex]. Substitute these values into the midpoint equation and solve for [tex]x_{b}[/tex] and [tex]y_{b}[/tex]:
[tex](1/2)\, (4 + x_{b}) = (-3)[/tex].
[tex]x_{b} = (-10)[/tex]
[tex](1/2)\, (1 + y_{b}) = (-3)[/tex].
[tex]y_{b} = (-7)[/tex].
Therefore, point [tex]B[/tex] will be at [tex](-10,\, -7)[/tex].
