To find the midpoint of a line segment with given endpoints, we use the midpoint formula. If we have two points [tex]\( P_1\left(x_1, y_1\right) \)[/tex] and [tex]\( P_2\left(x_2, y_2\right) \)[/tex], the midpoint [tex]\( M \)[/tex] is calculated as follows:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given the endpoints:
[tex]\[
P_1\left(\frac{1}{2}, \frac{3}{8}\right) \quad \text{and} \quad P_2\left(-\frac{9}{2}, \frac{5}{8}\right)
\][/tex]
First, we calculate the x-coordinate of the midpoint:
[tex]\[
x_{\text{mid}} = \frac{x_1 + x_2}{2} = \frac{\frac{1}{2} + \left(-\frac{9}{2}\right)}{2} = \frac{\frac{1}{2} - \frac{9}{2}}{2} = \frac{-\frac{8}{2}}{2} = \frac{-4}{2} = -2
\][/tex]
Next, we calculate the y-coordinate of the midpoint:
[tex]\[
y_{\text{mid}} = \frac{y_1 + y_2}{2} = \frac{\frac{3}{8} + \frac{5}{8}}{2} = \frac{\frac{8}{8}}{2} = \frac{1}{2} = 0.5
\][/tex]
Therefore, the midpoint of the line segment is:
[tex]\[
(-2.0, 0.5)
\][/tex]
So, the midpoint of the line segment with endpoints [tex]\( P_1\left(\frac{1}{2}, \frac{3}{8}\right) \)[/tex] and [tex]\( P_2\left(-\frac{9}{2}, \frac{5}{8}\right) \)[/tex] is [tex]\(\boxed{(-2.0, 0.5)}\)[/tex].
