How to Construct (Draw) a Perpendicular Line Through a Point
Perpendicular lines definition
Perpendicular lines are coplanar lines intersecting at 90° angles. By definition, two perpendicular lines create four 90° angles. Perpendicular lines are rare in nature, but humans use them in windows, at the four corners of doors, books, tables and in walls. Examine the mesh screens over your windows; all the little wires intersect as perpendicular lines.
You can construct perpendicular lines using only a straightedge, pencil, and drawing compass.
Constructing perpendicular lines
Perpendicular lines are easily constructed with high accuracy, whether you are an artist, mathematics student or architect. Begin by using a straightedge to draw a line. Be sure to label two points, such as A and E, near your arrowhead ends.
Label a point, perhaps P, roughly midway across your line segment. This is what you are given for the beginning of the construction:
Given: Point P on Line AE
Construct: Line SY perpendicular to AE at Point P
Construct a perpendicular line through a point on the line
Set the needle end of your drawing compass on Point P. Open the compass, so it reaches most of the way to the end of your drawn Line AE, for great accuracy. Swing the same distance on Line AE on either side of Point P.
These are just landing spots for the next step. Open the drawing compass up a lot more, and relocate the needle end of the drawing compass to one of the spots where the arcs cross Line AE.
Swing the compass above and below the Line AE, making sure you draw arcs that pass above and below Point P.
Without changing the distance on the drawing compass, relocate the needle end to the right-hand spot on Line AE where you first struck a tiny arc. Make sure not to change the setting on the drawing compass.
Swing the compass from that spot, above and below the line, so the two new arcs intersect the two most recent arcs.
Connecting the two arc intersections creates a line passing through Point P.
You could construct this with only arcs above OR below the line, but having both is a way to verify you have a perpendicular line through Point P. Label the perpendicular line SY, so now you have some SPY coming across some APE.
Construct a perpendicular line through a point off the line
Sometimes a point is above or below a given line, and you need a perpendicular line to pass through it. That is very easy to construct, too.
Given: Line OE with Point E above the line
Construct: Perpendicular Line ED
Follow these steps:
Place your drawing compass needle on Point E and open the compass enough that it reaches Line OE on both the left and right ends. Strike small arcs, so you mark the two places on Line OE.
Without changing the drawing compass, relocate the needle end on the left-hand arc where it intersects Line OE and swing an arc below Line OE.
Lift the drawing compass and, again without changing the distance on the compass, relocate the needle end to the other tiny arc where it crossed Line OE. Swing an arc below Line OE, so it crosses the other arc.
Use a straightedge to connect Point E to the intersection of the two arcs below the line. Call that Point D, and the point where your new line crosses OE can be Point N. Now you have ONE END to your task; you created Line ED perpendicular to Line OE, through Point E.
Proving perpendicular lines
Both these constructions can be proven perpendicular using the Linear Pair Perpendicular Theorem (congruent linear pairs must be 90°) or Euclid's Proposition 10, Bisecting a Straight Line.