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Perpendicular to a Line

Updated on July 24, 2016

In this hub I want to show an alternative way of constructing the perpendicular to a given line from an external point. This construction is very simple and it is done using basic construction techniques. The construction works due to Thales' theorem so I will go briefly over this theorem first.

Thales’ Theorem

So let's start by defining or describing Thales’ theorem. Looking at image 1, the theorem states the following: if A,B and C are points on a circle where the segment AC is a diameter of the circle, then the angle ABC is a right angle. In image 1 I included point O, which is the midpoint of AC or the center of the circle. I will not provide the proof of Thales’ theorem but as a hint you can draw segment OB, and this should create 2 isosceles triangles. If you are too lazy to continue the proof you can go to this Wikipedia page.

Thales' Theorem

Image 1
Image 1 | Source

Perpendicular to a given line from an external point

I don’t believe I saw this method of constructing the perpendicular to a line anywhere else, but this method is just as simple as the standard method taught in geometry classes. We should start with a line l and an external point P. We begin by marking a second point A, and this point A must be on line l. The second step is to draw segment AP. The third step is to find the midpoint M of segment AP. Point M can we found using basic geometric techniques like constructing the perpendicular bisector. The fourth step is to draw the circle with center at M and radius r=MA=MP. This circle will intersect line l at 1 or 2 points. If line l is a tangent to our circle, then the only point of intersection is point A. Our circle may intersect line l at a second point N. Thus we have 2 cases: 1) if line l is a tangent to the circle, then PA is perpendicular to l and 2) if the circle intersects line l at A and N, then PN is perpendicular to l. In image 2 I show the final drawing for a case where line l intersects the circle at two points.

Now if line l is tangent to the circle and the only point of contact is A, we know that l and PA are perpendicular since a tangent is perpendicular to the radius or diameter drawn to the point of contact. In the second case l is a secant and intersects the circle at A and N (image 2). The reason why PN is perpendicular to AN ,and by extension to line l, can be explained by Thales’ theorem. Points A, P and N are on the circle and segment AP is a diameter of the circle. Thus by Thales’ theorem the angle ANP=90 .

Perpendicular to a given line from an external point

Image 2
Image 2 | Source

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