

Perpendicular at a point on a line
Geometry construction using a compass and straightedge
This page shows how to draw a
perpendicular
at a point on a line with compass and straightedge or ruler. It works by effectively creating two
congruent triangles
and then drawing a line between their
vertices.
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
This construction works by effectively building two congruent triangles.
The image below is the final drawing above with the red lines added.

Argument 
Reason 
1 
Segment KP is congruent to KQ 
They were both drawn with the same compass width 
2 
Segment PR is congruent to QR 
They were both drawn with the same compass width 
3 
Triangles ∆KRP and ∆KRQ are
congruent 
Three sides congruent (sss). KR is common to both. 
4 
Angles PKR, QKR are congruent 
CPCTC. Corresponding parts of congruent triangles are congruent 
5 
Angles PKR QKR are both 90° 
They are a
linear pair
and (so add to 180°)
and congruent (so each must be 90°) 
 Q.E.D
Try it yourself
Click here for a printable construction worksheet containing two 'perpendiculars from 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
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