Topics
5. Rigid Transformations
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Geometry

5.04 Symmetry

Lesson
Practice

Line symmetry

Exploration

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

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For each type of line of reflection, try to drag the line to make the square and its image overlap.

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

A figure has line symmetry (sometimes called reflection 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 line symmetry.

This isosceles triangle has line 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 line 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 symmetry 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 other 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 symmetry for the square.

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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 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)

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The equations of the lines are:

  • y=x

  • y=4-x

  • x=2

  • y=2

Idea summary

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

Point symmetry

If a rotated shape perfectly overlaps the original shape after a rotation that is less than 360\degree, then the original shape has rotational symmetry. The point about which this rotation happens is called the center of rotation.

Exploration

Change the shape using the dropdown menu and click the 'Rotate 180 degrees' box to rotate the figure. Click 'Reset' to return the figure to its origianl position.

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  1. What do you notice as you rotate each shape 180\degree?

A specific rotational symmetry called point symmetry occurs when a figure is unchanged after a 180 \degree rotation.

Each point in a figure with point symmetry has a matching point the same distance from the central point, but on the opposite side.

This makes the central point of rotation a midpoint for each point and its image.

Notice that some figures have only point symmetry, or only line symmetry, but not both.

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While other figures have both line and point symmetry, or no symmetry at all.

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Examples

Example 3

For each figure, 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 point symmetry if it looks the same after being turned 180 \degree.

Apply the idea

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

A V-shape polygon with a vertical line of symmetry.
b
A parallelogram
Worked Solution
Create a strategy

A shape has line symmetry if half of the figure is a reflection of the other.

A shape has point symmetry if it looks the same upside down.

Apply the idea

The shape has no lines of symmetry.

The shape does have point symmetry because each point has a matching point on the opposite side of the center.

A parallelogram with points at two opposite vertices, at the mid of two opposite sides and at the center. The points are connected with line segments marked as congruent.
c
A quadrilateral
Worked Solution
Create a strategy

Determine if there is a line of symmetry that will create two equal halves of the shape.

Apply the idea

The shape has line symmetry.

A quadrilateral with a vertical line of symmetry
d
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 180 \degree rotation.

Apply the idea

This figure has neither line symmetry nor point symmetry.

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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 point symmetry.

A square inscribed in an equilateral octagon with vertical,horizontal and diagonal lines of symmetry.
A square inscribed in an equilateral octagon with points at the top and bottom vertices of the octagon, at the opposite vertices of the square and at the center. Line segments connecting the points are drawn and have congruent markings.
Reflect and check

Shapes with point symmetry are often radially symmetric. This creates shapes that have more than just point symmetry, such as having other types of rotational symmetry or line symmetry.

Example 4

Determine whether each of the following statements is true about the quadrilateral shown.

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y=0 is a line of symmetry.

Worked Solution
Create a strategy

A line of symmetry splits a shape into two equal halves. If the line y=0 divides the quadrilateral into two equal halves when drawn, it is a line of symmetry.

Apply the idea

The line y=0 is a horizontal line through 0 on the y-axis.

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The line splits the quadrilateral into two equal halves. The line y=0 is a line of symmetry.

Reflect and check

All corresponding points in the preimage and image (halves of the quadrilateral) are equidistant from the line of symmetry.

As an example, the distance from the line of symmetry to point A is 2 units, which is similar to the corresponding point B on the other side of the line of symmetry. Points C and D also have the same distance from the line of symmetry at 3 units.

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The quadrilateral has point symmetry.

Worked Solution
Create a strategy

A shape has point symmetry if it looks the same after being rotated 180 \degree.

Apply the idea

Remember a line is a straight angle measuring 180 \degree. If we think of each axis as a line we can rotate the points from one side of the axis to the other to create a 180 \degree rotation.

A quadrilateral on a cartesian plane. All four vertices are on the x and y axes. The rotation of the quadrilateral by 180 degrees is shown, with arrows indicating the initial and final points of each vertex.

The shape does not map back to itself. The quadrilateral does not have point symmetry.

Idea summary

A figure with point symmetry remains unchanged after a 180 \degree rotation.

The point of rotation for a figure with point symmetry acts as a midpoint for each point and its matching opposite.

Outcomes

G.RLT.3

The student will solve problems, including contextual problems, involving symmetry and transformation.

G.RLT.3a

Locate, count, and draw lines of symmetry given a figure, including figures in context.

G.RLT.3b

Determine whether a figure has point symmetry, line symmetry, both, or neither, including figures in context.

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