This lesson allows students to determine a process for constructing an equilateral triangle by finding the possible shapes within the construction of a regular hexagon. There is an opportunity to practice polygon vocabulary beyond equilateral triangles during the first activity. Students continue to practice straightedge and compass construction techniques as well as justify claims involving distance. Students make arguments and critique the reasoning of others when discussing claims about distance using circles (MP3).
One conjecture that builds toward subsequent lessons on proof via rigid motion is using rotation by 120 degrees to show that the equilateral triangle construction produces a triangle with all angles congruent and all sides congruent. Students are introduced to the word inscribed to describe a situation where a polygon sits inside a circle with all the vertices on the circle.
If students have ready access to digital materials in class, they can choose to perform all construction activities with the GeoGebra Construction tool accessible in the Math Tools or available at https://www.geogebra.org/m/VQ57WNyR.
Lesson overview
 4.1 Warmup: Notice and Wonder: Circles Circles Circles (5 minutes)

4.2 Activity: What Polygons Can You Find? (15 minutes)
 Digital applet in this activity

4.3 Activity: Spot the Equilaterals (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 4.4 Cooldown: I’m Stuck In A Circle! Help! (5 minutes)
Learning goals:
 Construct an equilateral triangle.
 Use circles in a construction to reason (using words and other representations) about lengths in figures.
Learning goals (student facing):
 Let’s identify what shapes are possible within the construction of a regular hexagon.
Learning targets (student facing):
 I can construct an equilateral triangle.
 I can identify congruent segments in figures and explain why they are congruent.
Required materials:
 Geometry toolkits
Glossary:

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.
 Access the complete Geometry Course glossary.
Standards:
 This lesson builds on the standard: CCSS.HSGCO.D.12MS.GCO.12MO.G.CO.D.11
 This lesson builds towards the standards: CCSS.HSGCO.A.3MS.GCO.3CCSS.HSGCO.CCCSS.HSGCO.D.12MS.GCO.12CCSS.HSGCO.D.13MS.GCO.13MO.G.CO.CMO.G.CO.A.3MO.G.CO.D.11
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