To find the midpoint of a segment [tex]\( P Q \)[/tex] with endpoints [tex]\( P = (3, 1) \)[/tex] and [tex]\( Q = (-3, -7) \)[/tex], we use the midpoint formula. The midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Here, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (3, 1) \)[/tex] and the coordinates of point [tex]\( Q \)[/tex] are [tex]\( (-3, -7) \)[/tex].
Substitute [tex]\( x_1 = 3 \)[/tex], [tex]\( y_1 = 1 \)[/tex], [tex]\( x_2 = -3 \)[/tex], and [tex]\( y_2 = -7 \)[/tex] into the midpoint formula:
[tex]\[
M_x = \frac{3 + (-3)}{2}
\][/tex]
[tex]\[
M_y = \frac{1 + (-7)}{2}
\][/tex]
Now, calculate each component separately:
1. For [tex]\( M_x \)[/tex]:
[tex]\[
M_x = \frac{3 - 3}{2} = \frac{0}{2} = 0
\][/tex]
2. For [tex]\( M_y \)[/tex]:
[tex]\[
M_y = \frac{1 - 7}{2} = \frac{-6}{2} = -3
\][/tex]
Therefore, the coordinates of the midpoint [tex]\( M \)[/tex] are:
[tex]\[
M = (0, -3)
\][/tex]
So, the midpoint of segment [tex]\( P Q \)[/tex] is [tex]\( (0, -3) \)[/tex].
