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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.
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Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions