Is it logical to conclude that the length (or distance) of a line segment is a negative number? Explain your reasoning.

Irrespective of which direction the line is pointing to, we always measure it ang obtain a positive value. And as for distance it's a fact that distance cannot be negative because it's a scalar quantity i.e. It only has magnitude, and no direction, this is why the measure of a line segment is always positive

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.