To find the midpoint of a segment, we use the midpoint formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Given the endpoints $\left(0, \frac{1}{8}\right)$ and $\left(-\frac{2}{5}, 0\right)$, we can substitute the coordinates into the formula:

$M = \left( \frac{0 + \left(-\frac{2}{5}\right)}{2}, \frac{\frac{1}{8} + 0}{2} \right)$

Simplifying the coordinates:

$M = \left( \frac{-\frac{2}{5}}{2}, \frac{\frac{1}{8}}{2} \right)$

$M = \left( -\frac{2}{10}, \frac{1}{16} \right)$

So, the midpoint is:

$M = \left( -\frac{1}{5}, \frac{1}{16} \right)$

The midpoint is $\left( -\frac{1}{5}, \frac{1}{16} \right)$.

Would you like more details or have any questions? Here are some related questions:

1. How do you find the midpoint of a line segment in 3D space?
2. What is the geometric meaning of the midpoint?
3. How does the midpoint formula change in different coordinate systems?
4. How can you apply the midpoint formula in real-life contexts?
5. What are other methods to verify the midpoint of a line segment?

Tip: Always simplify fractions before substituting into formulas to avoid mistakes.