In this activity, you will be investigating the symmetry found in regular polygons. A figure has reflection symmetry if you can reflect it across a line (called an axis of reflection or axis of symmetry) and get the same figure; that is, the image and pre-image coincide. It has rotational symmetry if you can rotate some number of degrees about some point and get the same shape in the same position. In this exploration, you will look for reflection and rotational symmetries of regular polygons. It is also good to note that the center of a regular polygon is defined to be the center of the circumscribed circle about the regular polygon.
Sketch 1
Step 1 : Construct any regular polygon and its interior: equlateral triangle, square, pentagon, hexagon, or other. (Construct it from scratch, or use a script.)
Step 2 : If the polygon's center does not already exist, construct it. (Find the center by constructing two medians of any two sides of your regular polygon. The intersection of the medians will be the center of the polygon.)
Step 3 : Construct a line and reflect your your figure's interior across the line. Give the image a light shade.
Investigation 1
Move the line until the image of your polygon coincides with the pre-image. (Be sure to move the line by dragging on the points on the line rather than the line itself.) When the image of a reflection is exactly the same as a pre-image, the reflection line is an axis of symmetry.Try the line in another position. How many reflection axes does your polygon have?
To look for rotation symmetries, select the center of the polygon and choose Mark Center in the Transform menu. Measure the angle. Rotate the figure by this angle, then manipulate the angle so that the rotated image fits exactly over the pre-image. What angle measure(s) cause(s) the figures to coincide? Repeat the exploration with other regular polygons.
Describe the reflection and rotation symmetries of the regular polygons. Be precise describing the lines that act as axes of symmetry, and state how many there are. List the angles that give the polygon rotational symmetry. Use your findings to make a conjecture about the reflectiona and rotational symmetries of a regular n-gon. Use a separate sheet of paper, if necessary.
Assignment - Day 8
1. Answer the questions posed in the investigations above.
2. Describe the new ideas you learned about today.
3. Discuss any questions or points of misunderstanding you may have had in the course of this lesson.
4. Use the conjectures you made in the sketches and investigations to answer each of the following concerning the reflection and rotational symmetries you found in your sketches:
a. equilateral triangle
b. square
c. pentagon
d. hexagon
e. n-gon
5. Construct each of the polygons described in questions 4a-4d using reflections and rotations, taking advantage of their symmetries.
6. Explore the symmetries in these other figures:
a. rhombuses
b. rectangles
c. isosceles trapezoids
d. kites.
Resources: Some of this material is taken from Exploring Geometry with The Geometer's Sketchpad, Blackline Activity Masters for Use with The Geometer's Sketchpad. Key Curriculum Press. 1996.