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.Other constructions pages on this site
Lines
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular at a point on a line
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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