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 step-by-step instructions

The above animation is available as a printable step-by-step 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.

Constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

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