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You are reading an older version of this FlexBook® textbook: CK-12 Middle School Math Concepts - Grade 7 Go to the latest version.

8.17: Lines of Symmetry

Difficulty Level: At Grade Created by: CK-12
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Have you ever seen an art exhibit made of nuts?

Kasey saw an exhibit made up of walnuts. Here is a picture of a walnut. Looking at this picture, can you tell if this walnut has line symmetry or rotational symmetry?

Do you know the difference?

This Concept will teach you all about the different types of symmetry. You will know how to answer these questions by the end of the Concept.

Guidance

We have mentioned that reflected figures are symmetrical, and that the line that acts as a mirror is called the line of symmetry.

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

Now let’s try another figure. Can we fold it perfectly in half?

We 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 we 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.

We 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 we turn it, one of the arms of the star always faces up.

The figure below, on the other hand, does not have rotational symmetry. Can you see why?

No matter how we work with this figure it will look different each time it is rotated. Therefore, we know that it does not have rotational symmetry.

Does each of the following figures have line symmetry, rotational symmetry, both, or neither?

Example A

Cross

Solution: Line Symmetry

Example B

Arrow

Solution: Neither

Example C

Quadrilateral

Solution: Line Symmetry

Here is the original problem once again.

Kasey saw an exhibit made up of walnuts. Here is a picture of a walnut. Looking at this picture, can you tell if this walnut has line symmetry or rotational symmetry?

Do you know the difference?

Line symmetry involves dividing a figure in halves. You can split a figure horizontally, vertically or on the diagonals and one half is a mirror image of the other half.

This walnut does have one vertical line of symmetry. You can only divide it in half vertically and have it create a mirror image.

The walnut does not have rotational symmetry. If you turn the walnut, then the view and image of the walnut is different.

The answer is line symmetry.

Vocabulary

Here are the vocabulary words in this Concept.

Rotation
a turn. The figure turns either clockwise or counterclockwise.
Line of Symmetry
the line that a figure reflects over. Also the line that divided a figure in half showing line symmetry.
Rotational Symmetry
the way a figure looks does not change no matter how you rotate it.
Symmetry
a figure that can be divided into two congruent halves is said to have symmetry.

Guided Practice

Here is one for you to try on your own.

Does this figure have line symmetry, rotational symmetry, both or neither.

Answer

This figure has line symmetry. Take a look.

When we can divide a figure or an object into two even matching halves, we say that the figure has line symmetry .

This figure can be divided in one way, vertically. If we tried to divide it horizontally, the two sides would not match.

Divided this way, the top half does not match the bottom half.

This figure does not rotate, but it does have line symmetry.

Video Review

Here is a video for review.

- This is a Khan Academy video on rotational symmetry.

Practice

Directions: 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.

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

Directions: Draw the second half of each figure.

Vocabulary

Bilateral Symmetry

Bilateral Symmetry

This occurs when a figure can only be divided into equal halves on one line. Such a figure has one line of symmetry.
Line Symmetry

Line Symmetry

A figure has line symmetry or reflection symmetry when it can be divided into equal halves that match.
Lines of Symmetry

Lines of Symmetry

Lines of symmetry are the lines that can be drawn to divide a figure into equal halves.
Reflection

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.
Rotational Symmetry

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.

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Difficulty Level:

At Grade

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Date Created:

Oct 29, 2012

Last Modified:

Apr 15, 2015
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