9.20: Lines of Symmetry
Have you ever built a half  pipe?
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 Concept is all about symmetry. Pay close attention and at the end you will know what the half pipe should look like.
Guidance
In geometry, we can look at a figure or an object and find the line symmetry in the figure or object. This Concept 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 a picture 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.
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.
Example A
Solution: One vertical line of symmetry
Example B
Solution: One horizontal line of symmetry
Example C
Solution: Three lines of symmetry from each vertex to the middle of the opposite side
Here is the original problem once again.
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 words in this Concept.
 Line Symmetry
 when a figure can be divided into equal halves that match.
 Bilateral Symmetry
 when a figure can only be divided into equal halve 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∘ around a fixed point looks exactly the same as it did in the beginning.
Guided Practice
Here is one for you to try on your own.
Determine the lines of symmetry in each figure.
Answer
The first figure, the cross has two lines of symmetry. One is vertical through the middle of the figure to divide it into even vertical halves. The other is horizontal through the middle of the figure to divide it into even horizontal halves.
The second figure, the arrow has one line of symmetry. It is a vertical line of symmetry.
The third figure does not have any lines of symmetry.
Practice
Directions: Identify the lines of symmetry in each figure or object. Draw them in if possible.
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Directions: Identify whether the following objects have rotational symmetry. If yes, write yes. If no, write no.
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Directions: Find three other objects. Draw them and identify each line of symmetry. Share your findings with a friend.
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
A figure has line symmetry or reflection symmetry when it can be divided into equal halves that match.Lines of Symmetry
Lines of symmetry are the lines 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.Rotational Symmetry
A figure has rotational symmetry if it can be rotated less than around its center point and look exactly the same as it did before the rotation.Image Attributions
Description
Learning Objectives
Here you'll learn to identify lines of symmetry.