To determine the coordinates of point [tex]\( B \)[/tex], let's use the fact that point [tex]\( M \)[/tex] is the midpoint of the line segment connecting points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. The midpoint [tex]\( M(x, y) \)[/tex] of a segment with endpoints [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] can be found using the midpoint formula:
[tex]\[
M_x = \frac{A_x + B_x}{2}, \quad M_y = \frac{A_y + B_y}{2}
\][/tex]
Given the coordinates of point [tex]\( A \)[/tex] as [tex]\( (-7, -9) \)[/tex] and the coordinates of the midpoint [tex]\( M \)[/tex] as [tex]\( (-0.5, -3) \)[/tex], we can set up the following system of equations:
[tex]\[
-0.5 = \frac{-7 + B_x}{2}
\][/tex]
and
[tex]\[
-3 = \frac{-9 + B_y}{2}
\][/tex]
First, solve for [tex]\( B_x \)[/tex]:
[tex]\[
-0.5 = \frac{-7 + B_x}{2}
\][/tex]
Multiply both sides of the equation by 2 to eliminate the fraction:
[tex]\[
-1 = -7 + B_x
\][/tex]
Add 7 to both sides of the equation to solve for [tex]\( B_x \)[/tex]:
[tex]\[
-1 + 7 = B_x
\][/tex]
[tex]\[
B_x = 6
\][/tex]
Next, solve for [tex]\( B_y \)[/tex]:
[tex]\[
-3 = \frac{-9 + B_y}{2}
\][/tex]
Multiply both sides of the equation by 2 to eliminate the fraction:
[tex]\[
-6 = -9 + B_y
\][/tex]
Add 9 to both sides of the equation to solve for [tex]\( B_y \)[/tex]:
[tex]\[
-6 + 9 = B_y
\][/tex]
[tex]\[
B_y = 3
\][/tex]
Thus, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (6, 3) \)[/tex].
