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Duplicating a Line Segment - Problem 3

Teacher/Instructor Brian McCall
Brian McCall

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

Using only one line segment, it is possible to create an equilateral triangle. First, draw a line segment. Then, duplicate this line segment using a compass. placing the end of the compass at one endpoint of the line segment, swing the compass in an arc. Repeat with the other endpoint of the line segment. Where these two lines intersect is the third point of the triangle. Using a straightedge to connect the points, you have drawn an equilateral triangle.

A common construction problem that you’ll see on a quiz or a homework is creating an equilateral triangle. Again you can only use a compass and a straight edge. And but creating an equilateral triangle, you’re also creating a 60 degree angle. So there’s two ways to answer this question. One you’re creating an equilateral triangle or if the question asks, 'Construct a 60 degree angle' you’re going to be doing that as well. But how are we going to do that?

Well, let’s start off by saying this is our side for all three sides of this triangle. We need a place to duplicate XY onto. So I’m going to draw a ray down below here. And onto this ray, I’m going to duplicate XY. So, I’ll start by drawing a ray and I’m going to say that this is going to be point XY, okay? So now I need to measure this distance. What is XY? So sharp end on one side and you’re going to expand your compass until it reaches to point Y. So I’m going to take the cap off and I’m going to make my mark on point Y. So that mark right there tells your teacher that you’ve measured that line segment.

So I’m going to go to point X and I’m going to swing a really big arc here. So I know that all of the points along this arc have this distance XY. Then I’m going to go over to this point right there which I’m going to make a dot at, and I’m going to do the same thing. I’m going to swing another arc. So I’m going to go over to this point and I’m going to swing another arc. So I know that this point right here has the length XY from both of these points.

So we’ve have just created an equilateral triangle, almost because we don’t have any side. So if your question asks creating a 60 degree angle, you only need to draw in one of these sides. So I’m going to draw that in right there. and if your question just says to construct a 60 degree angle you’re done. If it says to construct an equilateral triangle, you’re not going to draw a ray and you’re going to connect these last two points. So we’ve now created or constructed, since we only used a compass and a straightedge, an equilateral triangle.

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