Line symmetry

By Martin McBride, 2022-11-28
Tags: symmetry line symmetry reflective symmetry reflection
Categories: gcse geometry

We say that a shape is symmetrical if we can draw a line through the shape that divides it into two parts that are mirror images of each other. We call this type of symmetry line symmetry, or reflective symmetry.

We use the example of triangles to explore this further and then look at some other shapes.

Here is a video on the same topic:

Line symmetry of triangles

An isosceles triangle has two equal sides and two equal angles. This means that it is symmetrical:

Line symmetry of isosceles triangle

The dotted, vertical line divides the triangle into two halves, and each half is a mirror image of the other. If we were to fold the shape along the line, the two halves would match exactly.

The line goes from the top vertex of the triangle down to the centre of the base.

The dotted line is called a line of symmetry. There is only one way to fold an isosceles triangle, so we say it has one line of symmetry.

Here is a scalene triangle:

Line symmetry of scalene triangle

In a scalene triangle, all the sides and angles are different, so it is not possible to draw a line that divides the shape into two symmetrical halves. We say that the shape has no lines of symmetry.

Let's look at an equilateral triangle:

Line symmetry of equilateral triangle

In an equilateral triangle, all three sides and angles are equal. We can draw a line of symmetry from the top vertex to the centre base, similar to the isosceles triangle.

We can also draw lines of symmetry from the other two vertices to the opposite side.

An equilateral triangle, therefore, has 3 lines of symmetry. The triangle can be folded along any of the lines of symmetry and the two halves will match.

We normally draw all the lines of symmetry on a single diagram, like this:

Line symmetry of equilateral triangle

Line symmetry of a square

Here is a square:

Line symmetry of square

A square has a horizontal line of symmetry and a vertical line of symmetry, as shown on the left-hand square. It can be folded along either like to create a rectangle.

The two diagonals of a square are also lines of symmetry as shown on the right-hand square. It can be folded along either like to create a triangle.

This means that a square has four lines of symmetry. Here they are, all shown on the same diagram:

Line symmetry of square

Line symmetry of a rectangle

A rectangle has two lines of symmetry:

Line symmetry of rectangle

Unlike a square, a rectangle doesn't have diagonal lines of symmetry, as this diagram shows:

Line symmetry of rectangle

The diagonal divides a rectangle into two identical triangles, but they are not mirror images of each other. When the rectangle is folded over the diagonal line, the two parts do not match up, so the diagonal is not a line of symmetry.

Line symmetry of a trapezium

In general, the two legs of a trapezium do not have the same length or angle. This means that a trapezium has no lines of symmetry:

Line symmetry of trapezium

The exception is an isosceles trapezium. This is a trapezium where the two legs have equal length and have the same angle at the base (but both point inwards). In that case, there is one line of symmetry:

Line symmetry of trapezium

Line symmetry of a parallelogram

A parallelogram has no lines of symmetry.

Line symmetry of parallelogram

Line symmetry of a rhombus

A rhombus has two lines of symmetry, its diagonals:

Line symmetry of rhombus

Line symmetry of a kite

A kite has one line of symmetry:

Line symmetry of kite

Line symmetry of a regular polygon

A regular polygon with n sides has n lines of symmetry. However, the way the lines of symmetry are arranged depends on whether the number of sides is odd or even.

We have already seen a regular polygon with an odd number of sides. It is the equilateral triangle, which we looked at earlier. We saw that it has three lines of symmetry. Each line is drawn from a vertex to the middle of the opposite side.

Other regular polygons with odd numbers of sides follow a similar pattern:

Line symmetry of regular polygon

For example, the regular pentagon in the middle has 5 vertices and 5 sides. The lines of symmetry are drawn from each vertex to the midpoint of the opposite side.

Polygons with an even number of sides work differently. Again we have already seen an example, the square (a regular quadrilateral).

In the case of a square, there is a line of symmetry between the midpoint of the top and bottom sides. There is a line of symmetry between the midpoint of the left and right sides. The two diagonals are also lines of symmetry, making 4 in total.

Regular polygons with an even number of sides behave in a similar way to squares:

Line symmetry of regular polygon

In the case of a hexagon, for example, there are 3 pairs of opposite sides, with a line of symmetry that passes through the centres of each pair. There are also 3 pairs of opposite vertices, with a line of symmetry that passes through each pair.

This means that a hexagon has a total of six lines of symmetry, as you would expect for a six-sided regular polygon.

See also

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