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Constructing a Perpendicular to a Line - Problem 1
Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
To construct a perpendicular through a line segment from a point p that is not on the line, first create a line segment. This can be done by putting the sharp end of the compass at point p, and swinging an arc that goes through the line twice. This creates two points that can serve as endpoints to create a line segment. Since point p is the same distance away from both of these endpoints, it must be on the perpendicular bisector.
From one endpoint, draw an arc below the line, and do the same at the other endpoint. By connecting point p with the point where these two lines intersect, a line perpendicular to the original line is created.
An application of being able to bisect a line segment and create a perpendicular through a point on a line is this problem right here where you have been asked to construct a square where you know a perimeter.
Well let’s start off with the game plan. If this is the perimeter and we are constructing a square we don’t need to divide this into four congruent pieces. So the first thing that I’m going to do is bisect this line segment a, so that will give us two congruent pieces, so my second step is I’m going to have to bisect again and there we'll have our four congruent pieces.
So once I've done that then I’m going to duplicate one of those segments onto, and I guess segment is singular since we are talking about one, so I’m going to duplicate on to a new ray, I’m going to create a perpendicular through one of those endpoints and if I wanted to I could do that twice or I could just duplicate. So 5, I could say repeat. So let’s get started.
First thing we have to bisect this line segment twice. So I’m going to erase the a and got to extend my compass so it’s more than half of this line segment I’m going to swing an arc from each end.
Okay so now I know that those two points are on the perpendicularly bisector so we have found our midpoint of that segment but that’s not good enough because this is only half. So we're going to have to this again. We're going to make our compass opening a little smaller make sure that it’s more than a half and I’m going to swing an arc from the midpoint and from the endpoint that we started with and now I’m going to find my midpoint of my midpoint. So this will be ¼ of the total length of our segment. So I know this distance is going to be the length of every side of our square.
So let's come over here and we are going to have a ray onto each we are going to duplicate our square. So let’s measure this side length and duplicate it onto our ray. I guess to be technical if it’s a ray it’s going to have a fixed endpoint and that part is going to go on forever. So I’m going to duplicate that so I know that this is going to be one of my endpoints on my square.
So I know that a square has four congruent 90 degree angles. So I need to make
two perpendiculars through each of these points. So I’m going to extend the
ray a little bit in that direction and I’m using a different color and using
my compass I’m going to make another mark on this side. So I haven’t changed
it, so now I've made two marks here and I’m going to swing an arc down below. I'm going to move my ruler out of the way and that’s a little too much. So I’m
going to swing an arc down below like over here I’m going to swing another
arc, so now I’m going to have a perpendicular through this point.
So now we have one of our four 90 degree angles. So I guess what I could do is I could duplicate this 90 degree angle at this point over here. So I’m going to combine a couple of different techniques we’ve learnt in constructing so I’m going to swing an arc, I'm going to come over here and swing another arc and I’ll have to make that a little bit longer then I’m going to have to measure this distance with my compass and then make another mark to indicate that I have measured it and when I come up here I’m going to measure another one.
Now the other option I could have done is that I could have used this process over here to construct a perpendicular showing you a different way. So now I’m going to connect that point of intersection. So how do I know where how high up the sides go?
Well I need to go back to my original bisection and I need to re-measure that distance so I know the length of one side of my square, so I’m going to go to my original point and I’m going to make a mark right up here so I know that one point has to be right here.
I’m going to come over here and I know that another point has to be right here so I have the four corners of my square. Got one point right here, one point right here, one down here and our original point. So now I can connect these two and starting with the perimeter, which I bisected twice, I created a square by duplicating angles and constructing a perpendicular through a point.
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