Lesson 4

Construction Techniques 2: Equilateral Triangles

  • Let’s identify what shapes are possible within the construction of a regular hexagon.

4.1: Notice and Wonder: Circles Circles Circles

What do you notice? What do you wonder?

7 overlapping circles forming a 6 petaled flower in the center circle.

4.2: What Polygons Can You Find?

Here is a straightedge and compass construction of a regular hexagon inscribed in a circle just before the last step of drawing the sides:

 

  1. Use the polygon tool (the one that looks like a triangle) to draw at least 2 polygons on the figure. The vertices of your polygon should be intersection points in the figure. Shade in your polygons using different colors to make them easier to see. Use the style bar to change the color. This is what the style bar looks like.

    A circle, triangle and 3 partial horizontal stripes on a square.

  2. Write at least 2 conjectures about the polygons you made.

4.3: Spot the Equilaterals

Use straightedge and compass moves to construct at least 2 equilateral triangles of different sizes.



A figure composed of different shapes
An origami hexagonal coin purse.
  1. Examine the figure carefully. What different shapes is it composed of? Be specific.
  2. Figure out how to construct the figure with a compass and straightedge.
  3. Then, cut it out, and see if you can fold it up into a container like this.

Summary

The straightedge allows us to construct lines and line segments, and the compass allows us to make circles with a specific radius. With these tools, we can reason about distances to explain why certain shapes have certain properties. For example, when we construct a regular hexagon using circles of the same radius, we know all the sides have the same length because all the circles are the same size. The hexagon is called inscribed because it fits inside the circle and every vertex of the hexagon is on the circle.

Similarly, we could use the same construction to make an inscribed triangle. If we connect every other point around the center circle, it forms an equilateral triangle. We can conjecture that this triangle has 3 congruent sides and 3 congruent angles because the entire construction seems to stay exactly the same whenever it is rotated \(\frac{1}{3}\) of a full turn around the center.

Dashed circle with a point at center, 6 points equally spaced on the circle. Every other point is connected with a segment, creating an equilateral triangle.

Glossary Entries

  • circle

    A circle of radius \(r\) with center \(O\) is the set of all points that are a distance \(r\) units from \(O\). 

    To draw a circle of radius 3 and center \(O\), use a compass to draw all the points at a distance 3 from \(O\).

  • conjecture

    A reasonable guess that you are trying to either prove or disprove.

  • inscribed

    We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

     

  • line segment

    A set of points on a line with two endpoints.

  • parallel

    Two lines that don't intersect are called parallel. We can also call segments parallel if they extend into parallel lines.

  • perpendicular bisector

    The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it.