Perpendicular through a point

The question

Construct the perpendicular through P.

How it works.

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

01
Construct the perpendicular through P. Use only a compass and a straightedge — no measuring.
02
1. With the point on P, draw an arc that crosses the line twice. Both crossings are the same distance from P.
03
2. Call the crossings X and Y.
04
3. From X and then from Y, with the same opening, draw arcs that cross each other. This crossing is equally far from X and Y too.
05
4. Draw the line from P through that crossing. It meets the line at a right angle. Both P and the crossing are equally far from X and Y, so the line through them is the perpendicular bisector of XY — at right angles to the line.
06
Leave every arc showing — they show how the figure was made.