Topics
4. Rigid Motions
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United States of America
Grades 9-12

4.04 Symmetry

Lesson
Practice

Introduction

We reviewed reflections and rotations in lesson  4.02 Reflections  and lesson  4.03 Rotations  . These are transformations that lead to symmetry, through mirroring and circular motion. We will apply both types of symmetries here and determine whether shapes have line and rotational symmetry.

Reflection symmetry

Exploration

Change the line of reflection using the dropdown menu and drag the line of reflection.

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  1. What do you notice as you drag each line of reflection?

A figure has reflection symmetry (sometimes called line symmetry) if one half of the figure is the reflection of the other. This is equivalent to there being a line of reflection which maps a figure onto itself.

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We can see that the triangle's reflection across the line of reflection x=1 is just itself. This means that the triangle has reflection symmetry.

This isosceles triangle has reflection symmetry because of its characteristics. The segment lengths on either side of the line of symmetry are congruent, the line of symmetry is an angle bisector, and both base angles are congruent.

We can create a shape that has reflection symmetry using the reflection transformation.

A diagram showing an isosceles triangle reflected on its longer side is equal to a rhombus. The line of symmetry is drawn in the rhombus. Speak to your teacher for more details.

For example, when an obtuse isosceles triangle is reflected across its longest side, the shape created by combining the original triangle and the reflected triangle is a rhombus.

Since we built this rhombus using a reflection, we know that the rhombus is symmetric and has a line of reflection along its longest diagonal.

A diagram showing 2 asymmetric shapes. Speak to your teacher for more details.

A shape that has no lines of symmetry is called asymmetric.

Examples

Example 1

How many lines of symmetry does the following figure have?

A rhombus.
Worked Solution
Create a strategy

A line of symmetry is an imaginary line which, when drawn through the shape, makes one half of the shape a mirror reflection of the other half. We can draw suspected lines of symmetry and use tracing paper to verify that one half of the image maps onto the ohter half.

Apply the idea

There are two lines of symmetry in this figure.

A rhombus with its two lines of symmetry drawn. Speak to your teacher for more details.
Reflect and check

Notice that this figure has two lines of symmetry because all 4 of its side lengths are congruent and opposite angles are congruent.

Example 2

Determine the lines of reflection that map the square in the coordinate plane onto itself.

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Worked Solution
Create a strategy

Since it has four congruent sides and four congruent angles, a square has four lines of reflectional symmetry, two connecting the opposite corners and two connecting the midpoints of the opposite sides.

Apply the idea

The lines of symmetry for the square connect the following pairs of points:

  • \left(1,1\right) and \left(3,3\right)
  • \left(1,3\right) and \left(3,1\right)
  • \left(2,1\right) and \left(2,3\right)
  • \left(1,2\right) and \left(3,2\right)

which correspond to the lines

  • y=x
  • y=4-x
  • x=2
  • y=2
Reflect and check

If we sketch each of these four lines, we can see that the image of the square would be itself if we reflected across any of the lines. This confirms that they are lines of symmetry for the square.

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Idea summary

A figure has reflection symmetry if one half of the figure is the reflection of the other.

Rotational symmetry

Exploration

Change the shape using the dropdown menu and drag the slider to rotate the figure.

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  1. What do you notice as you drag the slider for each shape?

If a rotation maps a figure onto itself, then we say that the figure has rotational symmetry about that point. We can call that point the shape's center of rotation.

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For example, if we rotate the square about the origin by 90\degree counterclockwise, we can check that each point on the square will be mapped to another point on the square.

The result is that the image of the rotation will be the same as the pre-image.

We can see that the square also has rotational symmetry about its center for rotations of 180\degree and 270\degree as well. Every shape will match onto itself with a 360\degree rotation, so in order to say a shape has rotational symmetry, we must be able to map the pre-image to the image with less than a full rotation.

Similar to the case for line symmetry, we can make a shape that has rotational symmetry using the rotation transformation.

Speak to your teacher for the details of the image.

For example, we can start with a shape that has no rotational symmetry and make copies by rotating about one of its vertices. If we can fit a whole number of copies in a full rotation, then the shape that is made of all the copies will have rotational symmetry.

Examples

Example 3

Identify the rotation(s) that map ABCD onto itself.

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Worked Solution
Create a strategy
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If we look at the point of intersection of the axes of symmetry we can find the point of rotation for rotational symmetry.

Apply the idea
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Rotating about the point \left(1,2\right) by 90 \degree clockwise or 270 \degree counterclockwise, we can see that we do not get rotational symmetry.

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Rotating about the point \left(1,2\right) by 180 \degree clockwise or counterclockwise, we can see that we get rotational symmetry.

Rotating about the point \left(1,2\right), we can see that we only get rotational symmetry when we rotate by 180\degree clockwise about the point \left(1,2\right).

This is the function R_{180\degree,\left(1,2\right)}

Example 4

For each shape, determine the type(s) of symmetry present.

a
A figure that resembles the letter V. Speak to your teacher for more details.
Worked Solution
Create a strategy

A shape has line symmetry if a line can be drawn through it so that it divides the shape into two parts that are mirror images of each other.

A shape has rotational symmetry if it looks the same after being turned by some amount that is less than a full revolution.

Apply the idea

This shape has line symmetry, with a vertical line through the center.

b
A square.
Worked Solution
Create a strategy

Tracing the figure on tracing paper and performing reflections and rotations will help us decide the type of symmetry the figure has.

Apply the idea

The figure has both line symmetry and rotational symmetry.

c
A concave heptagon. Speak to your teacher for more details.
Worked Solution
Create a strategy

Determine whether a line can be drawn that can divide the shape into two mirror images of each other or if the shape could map to itself with a rotation less than 360 \degree.

Apply the idea

This figure has neither line symmetry nor rotational symmetry.

d
A square inscribed in an equilateral octagon. Speak to your teacher for more details.
Worked Solution
Apply the idea

The figure has both line symmetry and rotational symmetry.

Reflect and check

The figure has 4 lines of symmetry and rotational symmetry at 90 \degree, 180 \degree, and 270 \degree.

Idea summary

If a rotation less than 360 \degree maps a figure onto itself, then we say that the figure has rotational symmetry about that point.

Outcomes

G.CO.A.3

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G.CO.A.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. Graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

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