Lesson 15

Symmetry

  • Let’s describe some symmetries of shapes.

15.1: Back to the Start

Here is a segment \(AB\):
 

Line segment AB

If you translate the segment up 5 units then down 5 units, it looks the same as it did originally.

  1. What other rigid transformations create an image that fits exactly over the original segment?
  2. Are there any single rigid motions that do the same thing?

15.2: Self Reflection

Determine all the lines of symmetry for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:

  • the name of your shape
  • the definition of your shape
  • drawings of each line of symmetry
  • a description in words of each line of symmetry
  • one non-example in a different color (a description and drawing of a reflection not over a line of symmetry)


Look at all of the shapes the class explored and focus on those which had more than one line of symmetry.

  1. What is true for all the lines of symmetry in these shapes?
  2. Give an example of a shape that has two or more lines of symmetry that do not intersect at the same point.
  3. What would happen if you did a sequence of two different reflections across lines of symmetry for the shapes you explored in class?

15.3: Diabolic Diagonals

Kiran thinks both diagonals of a kite are lines of symmetry. Tyler thinks only 1 diagonal is a line of symmetry. Who is correct? Explain how you know.

Summary

A shape has symmetry if there is a rigid transformation which creates an image that fits exactly over the original shape. A shape has reflection symmetry if there is a reflection that takes the shape to itself, and the line of reflection in this case is called a line of symmetry. A regular hexagon has many lines of symmetry. Here are 2 of them. What other lines create a reflection where the image is the same as the original figure?

Two congruent hexagons, first with a line of symmetry through the midpoints of the top and bottom sides, second with line of symmetry through top right and bottom left vertices.

Glossary Entries

  • assertion

    A statement that you think is true but have not yet proved.

  • congruent

    One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.

  • directed line segment

    A line segment with an arrow at one end specifying a direction.

  • image

    If a transformation takes \(A\) to \(A'\), then \(A\) is the original and \(A'\) is the image.

  • line of symmetry

    A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.

    The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.

  • reflection

    A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line. 

    In the figure, \(A'\) is the image of \(A\) under the reflection across the line \(m\).

    Reflect $A$ across line $m$.

  • reflection symmetry

    A figure has reflection symmetry if there is a reflection that takes the figure to itself.

  • rigid transformation

    A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.

  • rotation

    A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.

    \(P'\) is the image of \(P\) after a counterclockwise rotation of  \(t^\circ\) using the point \(O\) as the center.

    ​​​​​Quadrilateral \(ABCD\) is rotated 120 degrees counterclockwise using the point \(D\) as the center.

  • symmetry

    A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).

  • theorem

    A statement that has been proved mathematically. 

  • translation

    A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.

    In the figure, \(A'\) is the image of \(A\) under the translation given by the directed line segment \(t\).