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 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

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.

Constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Tangents

Ellipses

Polygons

Non-Euclidean constructions

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