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An Overview of Point Symmetry

Chelsea Coons, Beverly Maitland-Frett
• Author
Chelsea Coons

Chelsea has taught middle school math and elementary school (all subjects) for 7 years. She has a Master's Degree in Educational Leadership and Administration and a Bachelor's Degree in Elementary Education with a concentration in Mathematics from Ball State University..

• Instructor
Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

Explore point symmetry. Learn the definition of point symmetry and understand how it differs from reflection. Find when point symmetry occurs with examples. Updated: 02/10/2022

What is Point Symmetry?

Point symmetry is when, given a central point on a shape or object, every point on the opposite sides is the same distance from the central point. Other terms for point symmetry include origin symmetry (origin is another word for the central point) and rotational symmetry. When viewed from opposite directions, the opposite sides or parts will look the same. To test if an object has point symmetry, rotate it on its central point or origin. If it has point symmetry, when rotated, it will match up with the other side.

In this rectangle, the central point or origin can be seen, labeled O. The shape has two diagonals demonstrating that the vertices are all equal distance from the origin.

What Is Symmetry?

Before we explore the definition of symmetry, let's complete an activity. Draw an uppercase X on a piece of paper. At the point where the lines cross, place a noticeable dot or point. What do you notice? Do you see two Vs, but one is upside down?

Now, draw an S. Is there a place on the S where you could place a point so that you have the same effect as with the X? If you chose right in the center of the S, then you are correct.

Point symmetry occurs when there exists a position or a central point on an object such that:

1. The central point splits the object or shape into two parts.
2. Every part on each has a matching part on the other that is the same distance from the central point.
3. Both parts face different directions.

Let's test our definition with the X and S. Notice the point splits both letters into two similar shapes, but they face different directions.

If you walk up to a mirror and touch the mirror with your finger, you would have made an example of point symmetry. Right where your finger touches the mirror is the point. It's as if you're connected to your image. That is the most important concept of point symmetry: there has to be a connection.

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• 0:01 What is Symmetry?
• 1:16 Point Symmetry vs Reflection
• 1:54 Examples of Point Symmetry
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When Does Point Symmetry Occur?

Point symmetry occurs in a variety of shapes and objects. In order for point symmetry to occur:

1. There must be a central point that divides the shape or object into two sides or parts, here referred to as Part A and Part B.
2. Every point on Part A must have a matching point on Part B that is the same distance away from the central point.
3. Part A and Part B are facing opposite directions.

Point Symmetry vs. Reflection

Line symmetry, or reflection, is when an object or shape has a line of symmetry that, if folded in half on this line, one side would match the other perfectly. Think of a person looking at their reflection in a pond. This is not the same as point symmetry. It is possible for a shape or object to have line symmetry, but not point symmetry. It is also possible for a shape or object to have both line symmetry and point symmetry. A shape can also have multiple lines of symmetry. The following are examples of line symmetry or reflection:

Line Symmetry Example One

This triangle has line symmetry because a line of symmetry can be drawn from one vertex to the midpoint of the opposite side. This line of symmetry would divide the shape perfectly in half. If folded along the line of symmetry, one side would match the other side perfectly. The triangle has three total lines of symmetry. It does not have point symmetry. If a central point was drawn in the middle, the points on one side would not all be the same distance from the central point as the points on the other side.

Line Symmetry Example Two

This star has line symmetry because if a line of symmetry is drawn down the center and folded in half on the line of symmetry, one side would match the other. It has five total lines of symmetry. It does not have point symmetry. If a central point was drawn, the points on one side would not be equal distance from the central point as the points on the other side.

Line Symmetry Example Three

This circle has line symmetry and point symmetry. If folded in half, both sides would match up. If rotated 180 degrees, the image would be the same.

Point Symmetry vs. Reflection

There is usually a misconception about symmetry and reflection. The difference is in the connection. It's as if you are connected to the image. If you stand three feet away from the mirror, you and your image are separate entities. However, with point symmetry, the object is not away from the image, it's onto the image.

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Video Transcript

What Is Symmetry?

Before we explore the definition of symmetry, let's complete an activity. Draw an uppercase X on a piece of paper. At the point where the lines cross, place a noticeable dot or point. What do you notice? Do you see two Vs, but one is upside down?

Now, draw an S. Is there a place on the S where you could place a point so that you have the same effect as with the X? If you chose right in the center of the S, then you are correct.

Point symmetry occurs when there exists a position or a central point on an object such that:

1. The central point splits the object or shape into two parts.
2. Every part on each has a matching part on the other that is the same distance from the central point.
3. Both parts face different directions.

Let's test our definition with the X and S. Notice the point splits both letters into two similar shapes, but they face different directions.

If you walk up to a mirror and touch the mirror with your finger, you would have made an example of point symmetry. Right where your finger touches the mirror is the point. It's as if you're connected to your image. That is the most important concept of point symmetry: there has to be a connection.

Point Symmetry vs. Reflection

There is usually a misconception about symmetry and reflection. The difference is in the connection. It's as if you are connected to the image. If you stand three feet away from the mirror, you and your image are separate entities. However, with point symmetry, the object is not away from the image, it's onto the image.

To unlock this lesson you must be a Study.com Member.

What is meant by point symmetry?

Point symmetry is when a shape or object has a central point that divides it into two sides. All points on one side are the same distance from the central point as the points on the other side.

What shapes have a point symmetry?

Any shape that has a central point that divides it into two sides, with all points on one side the same distance from all points on the other side. Some common examples are rectangles, squares, and circles.

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