Answer:
[tex]\sqrt{13}[/tex].
Step-by-step explanation:
Approach this question in the following steps:
By the midpoint formula, the midpoint of the segment between the point [tex](x_{0},\, y_{0})[/tex] and the point [tex](x_{1},\, y_{1})[/tex] would be:
[tex]\displaystyle \left(\frac{x_{0} + x_{1}}{2}, \frac{y_{0} + y_{1}}{2}\right)[/tex].
In this question, it is given that point [tex]{\sf M}[/tex] is the midpoint of the segment between point [tex]{\sf Q}[/tex] and point [tex]{\sf R}[/tex]. Hence, given the coordinates of point [tex]{\sf Q}[/tex] and [tex]{\sf R}[/tex], the coordinate of point [tex]{\sf M}[/tex] would be:
[tex]\displaystyle \left(\frac{(-3) + 1}{2}, \frac{7 + (-3)}{2}\right)[/tex].
Simplify to obtain: [tex](-1,\, 2)[/tex].
In a cartesian plane, the distance between the point [tex](x_{0},\, y_{0})[/tex] and the point [tex](x_{1},\, y_{1})[/tex] is:
[tex]\displaystyle \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2}}[/tex].
In this question, the goal is to find the length of segment [tex]{\sf PM}[/tex], which is the same as finding the distance between point [tex]{\sf M}[/tex] and point [tex]{\sf P}[/tex]. Since the coordinates of both points have been found, the length of this segment would be:
[tex]\begin{aligned} & \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2}} \\ =\; & \sqrt{(2 - (-1))^{2} + (4 - 2)^{2}}\\ =\; & \sqrt{13}\end{aligned}[/tex].