Constructing a Perpendicular Through a Point

The whole trick is turning a perpendicular problem into a bisector problem. Any point on the perpendicular bisector of a segment is equally distant from both endpoints. So if you can find two points on line m that are the same distance from your point P, then P itself sits on their perpendicular bisector — and that bisector is exactly the perpendicular line you want, already guaranteed to pass through P.

To get those two equidistant points, open your compass wider than the distance from P to the line, plant the needle on P, and swing one arc. Since P is off the line, that single arc crosses line m at two spots, call them X and Y. Both are the same distance from P because they both lie on the same arc. Now bisect segment XY the usual way: arcs from X and from Y, crossing above and below, connect those crossing points, and that line is your perpendicular — it passes through P and through the foot of the perpendicular on m.

When the point Q sits ON the line instead of off to the side, there's no outside point to swing an arc from in the same way, so you build your own two equidistant marks: center the compass at Q and swing an arc that cuts m on both sides at equal distance. From there the construction finishes exactly like before — bisect the segment between those two marks, and the bisector is the perpendicular at Q. This same bisection idea connects directly to constructing perpendicular bisectors of segments and to building right angles for other constructions.