# Lesson 4

Construction Techniques 2: Equilateral Triangles

### Lesson Narrative

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

### Learning Goals

Teacher Facing

• Construct an equilateral triangle.
• Use circles in a construction to reason (using words and other representations) about lengths in figures.

### Student Facing

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

### Student Facing

• I can construct an equilateral triangle.
• I can identify congruent segments in figures and explain why they are congruent.

Building On

Building Towards

### 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.