Perpendicular through a point, step by step

Drop a perpendicular from a point to a line.

The question

Construct the perpendicular through P.

compass+straightedge

Step 1→With the point on P, draw an arc that crosses the line twice

Step 2→Call the crossings X and Y

Step 3→From X and then from Y, with the same opening, draw arcs that cross each other

Step 4→Draw the line from P through that crossing. It meets the line at a right angle

arcs=the evidence

Construct the perpendicular through P. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
1. With the point on P, draw an arc that crosses the line twice. The arc from P cuts the line twice, and both crossings are the same distance from P. That symmetry is what the rest of the construction uses.
2. Call the crossings X and Y. Naming them X and Y makes the next step readable: you now have a segment XY whose midpoint is what you are heading for.
3. From X and then from Y, with the same opening, draw arcs that cross each other. Equal arcs from X and from Y cross at a point equally distant from both — the same fact the perpendicular bisector uses.
4. Draw the line from P through that crossing. It meets the line at a right angle. P is also equally distant from X and Y, so the line through P and that crossing is the perpendicular bisector of XY, and it meets the line at a right angle.
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.

How it works.

01
Construct the perpendicular through P. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
02
1. With the point on P, draw an arc that crosses the line twice. The arc from P cuts the line twice, and both crossings are the same distance from P. That symmetry is what the rest of the construction uses.
03
2. Call the crossings X and Y. Naming them X and Y makes the next step readable: you now have a segment XY whose midpoint is what you are heading for.
04
3. From X and then from Y, with the same opening, draw arcs that cross each other. Equal arcs from X and from Y cross at a point equally distant from both — the same fact the perpendicular bisector uses.
05
4. Draw the line from P through that crossing. It meets the line at a right angle. P is also equally distant from X and Y, so the line through P and that crossing is the perpendicular bisector of XY, and it meets the line at a right angle.
06
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.