Interactive · 1 question
Ok, we have a side, a compass, and a straightedge. We also need to construct a regular hexagon. How are we going to do this?
Well, what's do we know about a regular hexagon that can help us in this construction?
Let's take a look at a hexagon made of these triangles.
Cool! All of these lines are of the same length (since the triangles are equilateral)! So, we have 6 lines of the same length extending from a center. What does that make us think of?
A CIRCLE!!! Let's let O be the center of our circle. Our new diagram is:
What's more, all of the sides of the hexagon have the same length as the radius of the circle. Ok, let's get back to our CURRENT construction, which is a side. What should we do? Build an equilateral triangle off of it!
Right! Now what? We have an equilateral triangle, but what next?
We build a circle, with O as the center, just like we did in the diagram with the hexagon.
Next, we remember that the points on the hexagon are equally spaced around the circle, so we can just make a circle around A until it intersects the original circle, and that's vertex C.
We keep doing that, and eventually we find all of the points on the hexagon, and then we connect the dots.
Construction Summary
First, we make an equilateral triangle out of the side we're given.
Then, we make a circle around one of the points.
Next, we swing our compass around one of the OTHER points until it hits the first circle we made.
Then, we swing the compass around the point we just made (the intersection of the circles) until it hits the ORIGINAL circle, and we keep doing that until we get 6 points on the original circle, and that's our hexagon.