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Perpendicular at the endpoint of a ray
Geometry construction using a compass and straightedge

This page shows how to construct a perpendicular at the end of a ray with compass and straightedge or ruler. This construction works as a result of Thales Theorem. From this theorem, we know that a diameter of a circle always subtends a right angle to any point on the circle, so by using it in reverse we produce a right angle.

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.

Why it works

This construction works as a result of Thales Theorem. From this theorem, we know that a diameter of a circle always subtends a right angle to any point on the circle. In this construction we:
  • Create a circle that has the end part of the given ray as a chord. (step 3).
  • We then draw a diameter of the circle (step 4).
  • When we close the triangle in step 5, we know it must be a right triangle from Thales Theorem.

Proof

The image below is the final drawing from the animation above.

  Argument Reason
1 AB is the diameter of the circle center D. A diameter is a line through the center of a circle.
2 m∠APB = 90° From Thales Theorem - the diameter of a circle subtends a right angle to any point of the circle's circumference - here P.
3 AP is perpendicular to the endpoint of the ray PB From (2)

  - Q.E.D

Try it yourself

Click here for a printable worksheet containing two 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