Think of the red segment as part of another secant perpendicular to the first secant, as in the diagram below:
You know the radius of the circle ($R$ in the figure),
the length of the secant ($2L$ in the figure), and the distance from
the center of the secant to the point where the red segment intersects
the secant ($m$ in the figure).
Knowing $R$ and $L,$ you can use the Pythagorean Theorem on the right
triangle in the left half of the circle to find $h,$
the distance from the center of the circle to your secant line.
Notice that the thing that looks like a rectangle in the upper right quadrant of the circle actually is a rectangle: its edges are formed by two perpendicular secant lines above and on the right, and on the left and below by the radial lines perpendicular to those secant lines.
So the length of the right edge of that rectangle is $h,$ just like the left edge that you computed in the previous step.
Also, the bottom edge has the same length as the top edge, $m.$
Knowing $R$ and $m,$ you can use the Pythagorean Theorem on the right
triangle in the bottom half of the circle to find $k.$
But the segment labeled $k$ is only the bottom half of the secant in the right half of the circle. The top half of that secant, which is also of length $k,$ is made up of one edge of the rectangle (of length $h$)
and the red segment (of unknown length $x$).
Therefore $k = h + x.$ Solve for $x.$