Geometric constructions: perpendicular line through a point off the line | Geometry | Khan Academy

What I have here is a line, and I have a point that is not on that line. 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. So, what I will do is I'll pick an arbitrary point on our original line, let's say this point right over here. 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.

Now I can 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. I can do that by centering my compass on that offline point and then drawing another arc.

I can see very clearly that this point also has the same distance from this point up here. 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. What I can do next 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've made. 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, same as this one right over here, which is the same as this distance. Let me make sure I get 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.