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8.17: Lines of Symmetry

Difficulty Level: At Grade Created by: CK-12
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Cassie works at an apple orchard. She mostly works in the store and sells the products, but she loves going to the orchard and picking fresh apples off the trees. She has noticed that most, but not all of the apples are symmetrical. She cuts one apple in half and notices that the symmetry continues on the inside. Does the apple have line symmetry, rotational symmetry, both or neither?

In this concept, you will learn about lines of symmetry.

Lines of Symmetry

A line that acts as a mirror is called a line of symmetrySymmetry means that when you divide a figure in half, the halves are congruent. In other words, a figure is symmetric if its outlines mirror each other.

Look at the figure below. Imagine you can fold it in half. When you fold it, do the outlines of each half match? They do, so this figure has symmetry.

When you “unfold” the figure, you have two congruent halves. The “fold” line is the line of symmetry. It divides the figure into halves that are mirror images of each other. Every part on one half “mirrors” or corresponds to a part on the other half.

Let's look at an example. Can you fold the figure perfectly in half?

You can try to fold this figure a bunch of different ways, but it does not have a line of symmetry.

Rotational symmetry is a different kind of symmetry. It means that when you rotate a figure, the figure appears to stay the same. The outlines do not change even as the figure turns. Look at the figure below.

You can tell the figure has been rotated because the dot moves clockwise. However, the outlines of the figure have not changed. This figure has rotational symmetry because every time you turn it, one of the arms of the star always faces up.

Examples

Example 1

Earlier, you were given a problem about Cassie and the apples.

Look at the apple that she cut in half. Does the apple have line symmetry, rotational symmetry, both or neither?

First, determine if the shape can be folded in half over a line.

Yes.

Next, determine if the shape will look the same when it is rotated.

No.

Then, state the type of symmetry that the figure has.

Line symmetry.

The answer is that the figure has line symmetry.

Example 2

Does the figure below have rotational symmetry?

First, note the appearance of the figure as it turns.

The figure has a different appearance for each turn.

Next, remember the definition of rotational symmetry.

The figure appears to stay the same.

Then, state whether the figure has rotational symmetry.

No.


The answer is that the figure does not have rotational symmetry.

Example 3

Does the following figure have line symmetry, rotational symmetry, both, or neither?

First, determine if the shape can be folded in half over a line.

Yes.

Next, determine if the shape will look the same when it is rotated.

Yes.

Then, state the type of symmetry that the figure has.

Line symmetry and rotational symmetry.

The answer is that the figure has both line symmetry and rotational symmetry.

Example 4

First, determine if the shape can be folded in half over a line.

Yes.

Next, determine if the shape will look the same when it is rotated.

No.

Then, state the type of symmetry that the figure has.

Line symmetry.

The answer is that the figure has line symmetry.

Example 5

First, determine if the shape can be folded in half over a line.

Yes.

Next, determine if the shape will look the same when it is rotated.

No.

Then, state the type of symmetry that the figure has.

Line symmetry.

The answer is that the figure has line symmetry.

Review

Answer each of the following questions true or false.

  1. A reflection has rotational symmetry.
  2. A square has line symmetry and rotational symmetry.
  3. If a figure is a transformation than it always rotates clockwise.
  4. A slide is also called a translation.
  5. A flip has a line of symmetry because it is a reflection.
  6. A rotation or turn always moves clockwise and never counterclockwise.
  7. A star has rotational and line symmetry.
  8. An regular octagon has rotational and line symmetry.

Tell whether the figures below have line symmetry, rotational symmetry, both, or neither.

Draw the second half of each figure, and then rotate the figure.

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.17.

Resources

 

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Vocabulary

Line of Symmetry

A line of symmetry is a line that can be drawn to divide a figure into equal halves.

Reflection

A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.

Rotation

A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.

Rotational Symmetry

A figure has rotational symmetry if it can be rotated less than 360^\circ around its center point and look exactly the same as it did before the rotation.

Symmetry

A figure has symmetry if it can be transformed and still look the same.

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Difficulty Level:
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Date Created:
Dec 02, 2015
Last Modified:
Sep 08, 2016
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