Lesson 6 Symmetries of Regular Polygons Solidify Understanding

Jump Start

You are running around a circular track. What fraction of a lap around the track have you run if the angle formed by your starting position, the center of the track, and your current position measures:

1.

$180 °$

2.

$45 °$

3.

$270 °$

4.

$36 °$

5.

$72 °$

Learning Focus

Find patterns of line and rotational symmetry in regular polygons.

What makes a polygon regular or symmetric?

How does the symmetry of regular polygons differ depending on the number of sides?

What patterns can I find in the number and characteristics of the lines of symmetry in a regular $n$ -gon (a polygon with $n$ sides)?

What patterns can I find that describe the nature of the rotational symmetry in a regular $n$ -gon?

Open Up the Math: Launch, Explore, Discuss

A line that reflects a figure onto itself is called a line of symmetry. A figure that can be carried onto itself by a rotation is said to have rotational symmetry. A diagonal of a polygon is any line segment that connects non-consecutive vertices of the polygon.

For each of the following regular polygons, describe the rotations and reflections that carry it onto itself (be as specific as possible in your descriptions, such as specifying the angle of rotation).

1.

An equilateral triangle

2.

A square

3.

A regular pentagon

4.

A regular hexagon

5.

A regular octagon

6.

A regular nonagon

7.

What patterns do you notice in terms of the number and characteristics of the lines of symmetry in a regular polygon?

8.

What patterns do you notice in terms of the angles of rotation when describing the rotational symmetry in a regular polygon?

Ready for More?

You may have found a rule for the number of lines of symmetry numerically—by looking for patterns in an input-output table relating the number of sides of the polygon to the number of lines of symmetry. Can you justify why this conjecture is correct, since you haven’t tried all possible regular polygons?

Write a detailed argument to a friend explaining why there are $n$ lines of symmetry in a regular polygon, even though regular polygons with an odd number of sides have different types of lines of symmetry from those found in regular polygons with an even number of sides. Use precise and accurate language, and support your words with appropriate diagrams and illustrations.

Trade your paper with another student and read and critique each other’s explanations.

Takeaways

I conjectured that the number of lines of symmetry in a regular polygon with $n$ sides is

I can explain this by:

I conjectured that the smallest angle of rotation in a regular polygon with $n$ sides can be found by

This is because:

Vocabulary

bisect (verb); bisector (noun) (midpoint)
midpoint
n-gon
opposite angles, opposite vertices
opposite sides (in a parallelogram or an even-sided polygon)
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we examined lines of symmetry and rotational symmetry in regular polygons. We found that the number of lines of symmetry and the smallest angle of rotation could be related to the number of sides of the regular polygon.

Retrieval

1.

Find the coordinates of the vertices of the new quadrilateral formed if you reflect quadrilateral $A B C D$ over the given line.

Point $A^{'}$ :

Point $B^{'}$ :

Point $C^{'}$ :

Point $D^{'}$ :