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Constructing the Perpendicular Bisector - Problem 2
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
In order to create 5/4 of a line segment, first remember that 5/4 is 1 + 1/4 of a line segment. So, first find the length of 1/4 of the line segment, then add that to a copy of the original line segment.
To find 1/4 of a line segment, first find the midpoint of the line segment (recall that this can be done by drawing congruent arcs at each endpoint of the line segment, and drawing a line through the points where the arcs intersect). Then, repeat the procedure to find the midpoint of the new line segment, and the distance between one endpoint and the second midpoint found is 1/4 of the line segment.
An application of creating a perpendicular bisector or taking a line segment and dividing it in half is a problem kind of like this. We’ve been asked to construct 5/4 of a line segment that you’re given. So in this problem we’re given XY. But how do we come up with 5/4 of XY?
Well, we can write that 5/4 if we’re going back to fractions is equal to 1/4 plus 4/4. And 4/4 is just a whole. So if I copy, if I duplicate XY onto a new ray, and if I find a quarter of this line segment then if I add those up, I’ll end up with 5/4. So this will work no matter what kind of fractions you’re dealing with on your homework.
So let’s start of by creating our one whole of XY. So I’m going to grab my straightedge and I’m going to come over here and I’m gong to create a ray so that we can duplicate our line segment. So we’re going to make it nice and long, and this right here will be one endpoint of XY. So now what we need to do is we need to take our compass and we need to measure this distance. So all we’re doing right now is duplicating a line segment.
I’m going to come right here and I’m going to measure that and I’m going to come over here and I’m going to duplicate this line segment XY. So for keeping track on this equation here, we have created 4/4 or one whole of XY. But how do we come up with 1/4?
Well, if we start with a half, so let’s say we bisected this. And then let’s say we halved again. A half of a half is a quarter. So I see that if I bisect this line segment twice, we’ll come up with a quarter of XY. So grab your compass and we’re going to start off. I’m going to erase this equation because we know that we want to create one quarter and I’m going to swing an arc from one endpoint. So from X I’m going to swing an arc, and it kind of slid away from me there so I’m going to touch it up and I’m going to come over to Y and I’m going to swing that same arc.
So we come over to Y and I see that we have created two points that are equidistant from the end points so these must be on your perpendicular bisector. So what I’m going to do is I’m going to draw in over our midpoint. So I know at this point right here, this is half of XY. So if we bisect it again we’ll have a quarter.
So no time like the present, let’s grab our compass and we’re going to change its length a little bit, and I’m going to swing an arc from X and it move over to my midpoint and swing another arc using the same setting on your compass and now we have found our point that divides XY into one quarter. Are we done? No because we haven’t duplicated this one fourth into our new line.
So we’re going to measure this with our compass, sharp end on one of these endpoints. So I’m going to come here and I’m going to measure this and I’m going to make an arc. Keeping my compass the same, I’m going to come over to this end point which I know is the endpoint of XY and I’m going to swing another arc and we have now created from one endpoint to the next, this whole distance right here is 5/4 XY.
So the key step here is realizing that 5/4 is the same as one whole plus ¼ and then that we had to bisect this twice to find a fourth.
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