# 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

**, near your arrowhead ends.**

*E*Label a point, perhaps ** P**, roughly midway across your line segment. This is what you are given for the beginning of the construction:

**Given:**on*Point P**Line AE***Construct:***Line SY*at*AE**Point P*

### Construct a perpendicular line through a point on the line

Set the needle end of your drawing compass on

. Open the compass, so it reaches most of the way to the end of your drawn*Point P*, for great accuracy. Swing the same distance on*Line AE**Line AE**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

, making sure you draw arcs that pass above and below*Line AE**Point P**.*Without changing the distance on the drawing compass, relocate the needle end to the right-hand spot on

*Line AE**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

**, so now you have some**

*SY*

*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**Point E***Construct:**Perpendicular*Line ED*

**Follow these steps:**

Place your drawing compass needle on

*Point E**Line OE**Line OE**.**Without changing the drawing compass*, relocate the needle end on the left-hand arc where it intersects*Line OE**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

. Swing an arc below*Line OE*, so it crosses the other arc.*Line OE*Use a straightedge to connect

*Point E*, and the point where your new line crosses*Point D**OE*. Now you have*Point N**ONE END**Line ED*, through*Line OE**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.