Returning a Polygon to Its Original Position

The Lesson Activities will help you meet these educational goals:

  • Content Knowledge—You will describe the rotations and reflections that carry a given rectangle, parallelogram, trapezoid, or regular polygon onto itself.
  • STEM—You will grow in your understanding of mathematics as a creative human activity.
  • 21st Century Skills—You will employ online tools for research and analysis.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

  1. Rotations

A rotation is a transformation that can map a figure back onto itself. Use the GeoGebra geometry tool to explore this property of rotations and complete each step below.If you need help, follow these instructions for using GeoGebra.

  1. Use the slider to rotate pentagon ABCDE about the center of rotation, R. The angle through which the pentagon is rotated is CRC′, represented by the variable α, where 0° ≤ α≤ 360°. At what values of α does pentagon A′B′C′D′E′ coincide with pentagon ABCDE? Enter the values in the table. For each value of α that you enter, enter the vertex that coincides with vertex C′.

Sample answer:

α / C′
72° / B
144° / A
216° / E
288° / D
360° / C
  1. At how many values of α (0˚α ≤ 360˚) does pentagonA′B′C′D′E′ coincide with pentagon ABCDE? What is significant about this number?

Sample answer:

Each time the pentagon is rotated 72˚, it maps back onto itself. There are five values of α for which the pentagon maps back onto itself. This is the same as the number of sides of the pentagon.

  1. Now move the center of rotation, R, to a different position on the coordinate plane. Next, move the slider bar to rotate pentagon ABCDE. How many times does the pentagon map back onto itself in this situation? What does this tell you about the location of the center of rotation?

Sample answer:

If the center of rotation is changed, the pentagon maps back onto itself only one time, when the value of α is 360°—that is, when the pentagon completes a full rotation.

  1. Move the center of rotation back to the origin (0, 0) of the coordinate plane. Now alterpentagon ABCDE by moving one or more line segments on the pentagon to a different location. Rotate the pentagon using the slider. How many times does the pentagon map back onto itself in this situation? What can you conclude about the kinds of polygons that can map back onto themselves? Describe them.

Sample answer:

The changed pentagon maps back onto itself only one time, when the value of α is 360°. A polygon must be regular to map back onto itself more than once. In a regular polygon, all of the sides are equal and all of the angles are equal.

  1. List at least three shapes, other than pentagons, in the table. If the shape is a polygon, indicate whether the shape is regular or irregular. Find the number of times that will the shape will map back on to itself as the shape rotates 360°about its center. Also note how many degrees the shape has rotated each time it maps back onto itself. Use GeoGebra to guide you in this exercise, if you wish.

Sample answer:

Shape / Times Mapped onto Itself / Degrees Rotated
equilateral triangle / 3 / 120°
square / 4 / 90°
circle / infinite / infinite
  1. Reflections

A reflection is a transformation that can map a figure back onto itself. Use GeoGebra to explore this property of reflections and complete each step below.

  1. Hexagon AGHBGHCGHDGHEGHFGH is a reflected image of hexagon ABCDEF. The midpoints of the sides of hexagon ABCDEF are also shown. Drag the line of reflection, until the image coincides with the preimage. At this location, if the preimage flips about the line of reflection, it will flip onto itself. In how many different positions can you place so the image reflects onto the preimage in this manner? Describe the different positions. Be sure to pass the line of reflection through both vertices and midpoints before answering.

Sample answer:

HexagonAGHBGHCGHDGHEGHFGH coincides with hexagonABCDEF when passes through the midpoints of opposite sides; that is, it is a perpendicular bisector of the two sides. HexagonAGHBGHCGHDGHEGHFGH also coincides with hexagonABCDEF when the line of reflection joins a pair of vertices opposite one another on the hexagon. There are three perpendicular bisectors and three pairs of opposite vertices. In all, there are six lines of reflection that will map the hexagon back onto itself.

  1. You’ve seen how a polygon can reflect onto itself during a transformation. Use what you know about symmetry to describe the line of reflection required for such a transformation. Be sure to use the word symmetry in your answer.

Sample answer:

When the line of reflection coincides with a line of symmetry for a polygon or shape, then the polygon will be able to reflect onto itself.

  1. List at least three shapes, other than hexagons, in the table. If the shape is a polygon, indicate whether the shape is regular or irregular. Find the number of lines of reflection that will map each shape back on to itself when the shape flips about the reflection line. Also describe the position of the line of reflection in your own words. Use GeoGebra to guide you in this exercise, if you wish.

Sample answer:

Shape / Number of Lines of Reflection / Description
equilateral triangle / 3 / 3 perpendicular bisectors of the sides
regular heptagon / 7 / 7 perpendicular bisectors of the sides
regular octagon / 8 / 4 perpendicular bisectors and 4 lines joining pairs of opposite vertices
  1. Review your work in parts a through c. What pattern do you observe regarding the number of lines of reflection through a regular polygon that will map the polygon onto itself? How does this number relate to the number of lines of symmetry for the same polygon?

Sample answer:

The number of sides of a regular polygon is the same as the number of lines of reflection that can map the polygon onto itself. The number of lines of symmetry is the same as the number of lines of reflection for mapping.

  1. Some regular polygons have an even number of sides. Other regular polygons have an odd number of sides. How does the number of sides, even or odd, affect the position of the line of reflection for mapping the polygon onto itself? Explain. Use GeoGebra to guide you, if you wish.

Sample answer:

When a line of reflection coincides with a line of symmetry, the polygon maps onto itself. The position of the line of symmetry depends on the number of sides. For regular polygons with an even number of sides, lines of symmetry are the perpendicular bisectors of the opposite sides and the angle bisectors of opposite vertices. However, the symmetry of regular polygons with an odd number of sides does not allow lines of symmetry to connect the opposite pairs of vertices, so they have symmetry only along the perpendicular bisectors.

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