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# Point Symmetry|Definition & Meaning

## Definition

When all the **parts** have a **matching** part that is at the **same distance** but **opposite in direction** from **point 0**, it is called **point symmetry**. It can also be sometimes called the “**Rotational symmetry of Order 2.**”

In geometry, many **factors** are associated with symmetry. The concept of **symmetry** is utilized by professionals, artists, jewelry and clothes designers, architects, and many others for different jobs according to their **professions** and needs. There can be found **symmetrical motifs** in every place: flowers, beehives, handkerchiefs, religious symbols, and rugs.

## When Does Point Symmetry Occur?

In this figure, the **origin** or central point is **labeled O**. The shape consists of **two diagonals** showing the distances of all vertices from the origin are the **same**.

Point symmetry takes place in many kinds of **objects **and** shapes**. For point symmetry to happen, there must be a **dividing part **or** central point** that breaks the shape or object into two **equal parts** which are labeled here as **Part A **and** Part B**. Every point in Part A should have a matching point in Part B whose distance from the central point is just as same as the opposite point and lastly, the directions of Part A and Part B must be opposite.

You can physically **observe** point symmetry by looking in a mirror, and when you **touch** the mirror with your finger, where your finger touches the mirror is the **point**. It looks like you are inseparably hooked to your **image**.

An important rule in point symmetry is that there should be a point where they **connect** (the point where your finger is touching the mirror). An object is said to have point symmetry when you turn an object **upside-down** and it still looks the **same**. The figure and the matching part must face the **same way**.

### How to Test for Point Symmetry

To check if an object has point symmetry, **match** the object with the opposite side after **rotating** it on its **central point** or **origin**. if both objects **match** then, it has a **point symmetry.**

Point symmetry happens when we are given a central point on a shape and every point is at the same distance on both sides from the central point. **Different words** that can be utilized rather than point symmetry are **origin symmetry** (another word for the central point is the origin) and **rotational symmetry**. When looked at from opposite directions, the object will look exactly the same.

## Visual Examples of Point Symmetry

Point symmetry happens when there is a place or a point on an object from where the object can be divided into equal and identical parts. The **central point** divides the object or shape into two parts. Each part has a **connected** part on the other side whose distance from the central point is the same as the other. Both parts are placed in **opposite directions.**

When an object is made around a **single point**, it has point symmetry and that single point will also be the **point of symmetry**. This spot is known as the **object’s center** or the symmetry’s center. In the figure underneath, you can see a point marked **X′** on the opposite side of the center that reflects point **X** on the figure and is directly **inverse** to **X **and is on the figure. The figure is **symmetrical** around the center as proved above.

When after **rotating** a figure **180∘**, it comes back to its **original form**, we can say that it **has point symmetry**.

## Point Symmetry vs. Reflection

**Line symmetry,** or reflection, is an object having a **line of symmetry** and **not just a point** that, when **folded** on this line, one side would **match** the other side **completely**. When a person is having a look at their reflection on standing water. This is not the same as **point symmetry**. An object or shape **can** have **point symmetry** but **not** **line symmetry**. It is also possible for an object or shape to have **both line symmetry** and **point symmetry** at the same time. An object or shape might also have **multiple lines of symmetry**.

The triangle shown above has multiple lines of symmetry because a line of symmetry can be **drawn** from one **vertex** to the center point of the other side. This line of symmetry will **split** the shape completely in **half**. When **folded** from the line of symmetry, one side of the object would fit completely in comparison to the other side. **The triangle has three total lines of symmetry.** It does not have point symmetry.

## Point Symmetry in Letters

The symmetry point is a point that acts as a sort of **center point** for the object. If a line is drawn via the **point of symmetry** and it travels across the figure on one side of the point, then the line will also cross the diagram at a **similar distance** from the point on the **other side** of the point as well.

If all the parts of an object have a **matching part**, it will also have point symmetry. Point symmetry can also be seen in the **English alphabet**. The related sections of central point O are in opposing directions.

Point symmetry can be found in the following **capital letters:** **H, I, N, O, X, ****and O**. Point and line symmetry can be tracked down in the letters **H, I, O, and X.**

## Solved Examples: Point Symmetry

Which of the following letters of the English alphabet has a point symmetry yet doesn’t have a line of symmetry?

- H
- I
- Z
- X

### Solution

A line that cuts a figure into** two matching parts** is a **line of symmetry**.

For the letter Z, the point symmetry applies and is true, but for line symmetry, it does not apply. In comparison, both point and line symmetry can be applied to the letters H, I, O, and X.

Therefore, the correct answer is option **Z**.

*All images/mathematical drawings were created with GeoGebra.*