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9.7: Line Symmetry

Created by: CK-12

Introduction

Symmetrical Ramps

The plan is just about finished and the trio of students is feeling very good about their work. In the process of finishing the plan, Mr. Craven, the art teacher, took a look at their design. He saw a flaw in the design of the halfpipe that the three had drawn.

Mr. Craven told Isaac, Marc and Isabelle that their ramp was not symmetrical.

“If it isn’t symmetrical, it isn’t an accurate halfpipe,” Mr. Craven told them as he walked out of the room. “Let me know if you need a hand fixing it. You want to have it accurate before the presentation.”

Isaac looked at Marc and Marc looked at Isabelle, who shrugged. At that moment, Ms. Watson, the librarian, walked by.

“Why the long faces?” she asked. Then after seeing the plan, she said “Wow! that is some very fine work.”

“Yes, but Mr. Craven said that the half pipe isn’t symmetrical and it needs to be,” Isabelle explained.

“Oh, I see,” said Ms. Watson, looking again. “Well, that is easy enough to fix.”

How could a half pipe not be symmetrical? What is Mr. Craven talking about? This lesson is all about symmetry. Pay close attention and at the end you will know what the half pipe should look like.

What You Will Learn

By the end of this lesson you will be able to demonstrate the following skills:

  • Identify lines of symmetry in figures and objects
  • Draw figures with specified symmetry
  • Recognize congruence in mirror images

Teaching Time

I. Identify Lines of Symmetry in Figures and Objects

In geometry, we can look at a figure or an object and find the line symmetry in the figure or object. This lesson will teach you all about symmetry and about the different types of symmetry.

Here is a butterfly. Notice that we can draw a line right down the center of the butterfly and one side will match the other side. Here is what that looks like.

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.

Therefore, we can say that the butterfly has bilateral symmetry. Bilateral symmetry means that it has one line of symmetry that divides the butterfly in half.

What is a line of symmetry?

A line of symmetry is a line that splits a figure into symmetrical parts. In the butterfly, there is one line of symmetry that can be drawn to show the two equal matching halves of the butterfly.

Let’s look at an example of a figure that has more than one line of symmetry.

Here is a cross. This cross has four lines of symmetry. We can divide it vertically and horizontally and both sides will match. That shows two lines of symmetry.

Can you find the other two lines of symmetry in the cross?

Are there other types of symmetry?

Yes. There is turn or rotational symmetry. Rotational Symmetry means that you can rotate the figure around a fixed point and it will look the same.

Example

This star has rotational symmetry. It looks exactly the same no matter which point is rotated to be at the top. Since there are 5 points, this is a figure with a rotational symmetry of 5.

Try a few of these on your own. Identify the lines of symmetry in each object.

1.

2.

3.

Take a few minutes to check your work with a partner.

II. Draw Figure with Specified Symmetry

Now that you have learned about identifying lines of symmetry and about bilateral and rotational symmetry, it is time to draw some symmetrical figures on your own.

The first one to work with is bilateral symmetry. Remember that bilateral symmetry means that you can divide the object or figure into two matching sections. Here is an example.

Example

The letter B has bilateral symmetry. We can only divide it into two equal halves using one line of symmetry.

Now draw one of your own. Think about another letter that has bilateral symmetry and draw it on your paper.

Show a partner your work and explain how your figure has bilateral symmetry.

The next type of symmetry is rotational symmetry. Remember that rotational symmetry means that you can rotate the figure around a fixed point up to 360^\circ and it will still remain the same.

Example

Notice how the snowflake has rotational symmetry.

Now draw one of your own. Draw a figure or an object that has rotational symmetry.

Show your work to a partner and explain how your object has rotational symmetry.

III. Recognize Congruence in Mirror Images

We can look at mirror images of the same figure to recognize congruence. Remember that congruent means exactly the same size and shape. Now think about a mirror, it reflects something exactly the same, but opposite. The size and shape are the same even though the image may look backwards or opposite.

You can look at a mirror image and identify congruence in the image. Here is an example.

Example

This is a picture of a woman cleaning a mirror. Notice that her hands match up-the person is exactly the same, in fact everything is exactly the same or congruent from the real life image to the image in the picture.

Images reflected in a mirror are congruent in every way.

Let’s apply this to geometry.

Here one arrow is a mirror image of the other. But even though the figures are opposite, they are mirror images of each other and are therefore congruent.

Be sure that mirrored images are congruent. Sometimes they may appear congruent but are not. Looks can be deceiving!

Real Life Example Completed

Symmetrical Ramps

Reread the problem and underline any important information.

The plan is just about finished and the trio of students is feeling very good about their work. In the process of finishing the plan, Mr. Craven, the art teacher, took a look at their design. He saw a flaw in the design of the half pipe that the three had drawn.

Mr. Craven told Isaac, Marc and Isabelle that their ramp was not symmetrical.

“If it isn’t symmetrical, it isn’t an accurate half pipe,” Mr. Craven told them as he walked out of the room. “Let me know if you need a hand fixing it. You want to have it accurate before the presentation.”

Isaac looked at Marc and Marc looked at Isabelle, who shrugged. At that moment, Ms. Watson, the librarian, walked by.

“Why the long faces?” she asked. Then after seeing the plan, she said “Wow! that is some very fine work.”

“Yes, but Mr. Craven said that the half pipe isn’t symmetrical and it needs to be,” Isabelle explained.

“Oh, I see,” said Ms. Watson, looking again. “Well, that is easy enough to fix.”

How could a half pipe not be symmetrical? What is Mr. Craven talking about?

A half pipe has two halves to it. If it is not symmetrical it means that one half is a different size from the other half. Mr. Craven must have noticed that the drawing was inaccurate.

The students can fix the drawing by making sure that the measurements are accurate.

Vocabulary

Here are the vocabulary works that are found in this lesson.

Line Symmetry
when a figure can be divided into equal halves that match.
Bilateral Symmetry
when a figure can only be divided into equal halves on one line. This figure has one line of symmetry.
Lines of Symmetry
the lines that can be drawn to divide a figure into equal halves. Some figures have multiple lines of symmetry. Some figures have one line and some have no lines of symmetry.
Rotational Symmetry
When a figure rotated up to 360^\circ around a fixed point looks exactly the same as it did in the beginning.
Congruent
exactly the same size and shape

Technology Integration

  1. http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/4_Line_Symmetry/index.html – This is a GREAT video on symmetry and lines of symmetry.
  2. http://www.mathexpression.com/rotational-symmetry.html#GoToVideo – This is a quick video on rotational symmetry.

Time to Practice

Directions: Identify the lines of symmetry in each figure or object. Draw them in if possible.

1.

2.

3.

4.

5.

6.

Directions: Identify whether the following objects have rotational symmetry. If yes, write yes. If no, write no.

7.

8.

9.

10.

11.

12.

Directions: Draw your own objects/figures.

13. Draw a figure that has bilateral symmetry.

14. Draw a figure that has rotational symmetry.

15. Draw a figure that has two lines of symmetry.

16. Draw a figure that has rotational symmetry.

17. Draw a figure that has multiple lines of symmetry.

18. Draw a heart and identify the lines of symmetry.

19. Draw the letter H and identify the lines of symmetry.

20. Draw the number 8 and identify the lines of symmetry.  

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