To find the coordinates of point [tex]\( B \)[/tex], given that point [tex]\( M \)[/tex] is the midpoint of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we will use the midpoint formula.
The midpoint formula states that the coordinates of the midpoint [tex]\( M \)[/tex] of a segment with endpoints [tex]\( A(x_A, y_A) \)[/tex] and [tex]\( B(x_B, y_B) \)[/tex] are given by:
[tex]\[
M\left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right)
\][/tex]
Given:
- Point [tex]\( A \)[/tex] is at [tex]\( (-3, -5) \)[/tex]
- Point [tex]\( M \)[/tex] is at [tex]\( (-1, -7) \)[/tex]
Midpoint formula for [tex]\( x \)[/tex] coordinates:
[tex]\[
-1 = \frac{-3 + x_B}{2}
\][/tex]
Solving for [tex]\( x_B \)[/tex]:
1. Multiply both sides by 2 to get rid of the fraction:
[tex]\[
-2 = -3 + x_B
\][/tex]
2. Add 3 to both sides to isolate [tex]\( x_B \)[/tex]:
[tex]\[
x_B = 1
\][/tex]
Midpoint formula for [tex]\( y \)[/tex] coordinates:
[tex]\[
-7 = \frac{-5 + y_B}{2}
\][/tex]
Solving for [tex]\( y_B \)[/tex]:
1. Multiply both sides by 2 to get rid of the fraction:
[tex]\[
-14 = -5 + y_B
\][/tex]
2. Add 5 to both sides to isolate [tex]\( y_B \)[/tex]:
[tex]\[
y_B = -9
\][/tex]
Therefore, the coordinates of point [tex]\( B \)[/tex] are:
[tex]\[
(1, -9)
\][/tex]
