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## High school geometry

### Course: High school geometry>Unit 3

Lesson 8: Constructing lines & angles

# Geometric constructions: perpendicular line through a point not on the line

We can construct a perpendicular line through a point off the line. The lines must be perpendicular because a rhombus's diagonals are perpendicular.

## Video transcript

- [Instructor] What I have here is a line, and I have a point that is not on that line, and my goal is to draw a new line that goes through this point, and is perpendicular to my original line. How do I do that? Well you might imagine that our compass will come in handy, it's been handy before. And so what I will do is, I'll pick an arbitrary point on our original line, let's say this point right over here. And then I'll adjust my compass, so the distance between the pivot point and my pencil tip, is the same as the distance between those two points. And then I can now use my compass to trace out an arc of that radius. So there you go. Now my next step is to find another point on my original line that has the same distance from that point that is off the line. And I can do that by centering my compass on that offline point, and then drawing another arc. And I can see very clearly that this point also has the same distance from this point up here. And then I can center my compass on that point, and notice I haven't changed the radius of my compass, to draw another arc like this, to draw another arc like this. And then what I can do is connect this point and that point, and it at least looks perpendicular, but we're going to prove to ourselves that it is indeed perpendicular to our original line. So let me just draw it, so you have that like that. So how do we feel good that this new line that I just drew is perpendicular to our original one? Well let's connect the dots that we made. So if we connect all the dots we're going to get a rhombus. We know that this distance, this distance is the same as this distance. The same as this one right over here, which is the same as this distance, so let me make sure I got my straight edge right. Same as that distance, which is the same as this distance, same as that distance. And then, so this is a rhombus, and we know that the diagonals of a rhombus intersect at right angles. So there you have it. I have drawn a new line that goes through that offline point, and is perpendicular to our original line.