which step is the same in the construction of a perpendicular line through a point on a line and the construction of a perpendicular line through a point off a line?

In both cases the first step is the same. They start placing a compass on point P (which is on or off the line) and swinging an arc of any size that crosses the line twice. This creates a segment on the line that can be bisected.

Then, the steps differs for each case. In the case that the point P is on the line, a larger arc than the previous drawn is needed to be drawn from the extremes of the segment made before. In the other case, the same arc as the previous drawn is used.

The final step is also the same. This consist in connecting the intersection of the two small arcs made to point P.

Place your compass point at P and draw an arc of any radius below the line that crosses the line twice.

Step-by-step explanation:

In both constructions, the first step is the same. First place your compass point at P, and draw an arc of any radius below the line that crosses the line twice. Then proceed with other steps to complete each construction.