

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.
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Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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