
Perpendicular at the endpoint of a ray
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 stepbystep instructions
The above animation is available as a
printable stepbystep 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
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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