

Perpendicular to a line from an external point
Geometry construction using a compass and straightedge
This page shows how to construct a
perpendicular
to a line through an external point, using only a compass and straightedge or ruler. It works by creating a
line segment
on the given line, then
bisecting it. The bisector
will be a right angles to the given line. (See proof below).
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Proof
The image below is the final drawing above with the red lines added.

Argument 
Reason 
1 
Segment RP is congruent to RQ 
They were both drawn with the same compass width 
2 
Segment SQ is congruent SP 
They were both drawn with the same compass width 
3 
Triangle RQS is congruent to triangle RPS 
Three sides congruent (sss), RS is common to both. 
4 
Angle JRQ is congruent to JRP 
CPCTC. Corresponding parts of congruent triangles are congruent. 
5 
Triangle RJQ is congruent to triangle RJP 
Two sides and included angle congruent (SAS), RJ is common to both. 
6 
Angle RJP and RJQ are congruent 
CPCTC. Corresponding parts of congruent triangles are congruent. 
7 
Angle RJP and RJQ are 90° 
They are congruent and supplementary (add to 180°). 
 Q.E.D
Try it yourself
Click here for a printable construction worksheet containing two 'perpendiculars through a point' problems to try.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
(C) 2009 Copyright Math Open Reference. All rights reserved

