What Is Point Symmetry Definition Formula Properties and Solved Examples
In mathematics refers to a specific pattern in which we may draw a centre dividing line on a form or object and it offers two perfectly identical sides termed symmetry, and the shape or thing is referred to as symmetrical when this occurs. Examples of symmetrical figures are butterflies, and honeycombs made by bees. There are different types of symmetry, one of them being point symmetry. Let us now learn about it.
What Is Point Symmetry?
Point symmetry occurs when every part has a mirror portion. It is the same distance as the centre point but in the opposite direction. It is sometimes referred to as "Order 2 rotational symmetry." However, the two conditions listed below can be used to see if a shape possesses symmetry around a point.
Every portion of the defined form must be the same distance from the centre as the others.
The part of the form and its equivalent part must be in opposing directions.
What Is Line Of Symmetry?
The line of symmetry is an imaginary line that divides a form into two precise or identical halves. This line of symmetry in math shows the symmetry in form along only one axis. Many forms can have many lines of symmetry because they can be split into half by more than one line. A location within an object or figure through which any straight line passes, as well as two points on the figure's edge at the same distance from the centre but on opposing sides, is the centre of symmetry.
The following are the attributes of the line of symmetry:
It always travels through the shape's centre.
The halves formed by creating the symmetry line reflect each other. As a result, this sort of symmetry is also known as reflection symmetry.
Because the halves formed by drawing the line of symmetry math are mirror copies of each other, the symmetry is also known as mirror symmetry.
Point Symmetry Is Seen In Alphabet
When you draw a line across the point of symmetry, it will intersect the figure such that one side of the point would be at the same distance from the point. Point symmetry exists when every piece of an item has an equal part and can be seen in the English alphabet. The alphabet O has the centre point, and the equivalent parts are oriented in opposite directions. The uppercase letters H, I, N, O, X and Z all have point symmetry; when you draw a line from the middle of H, I, O and X you will get both upper and lower parts of the same length. The letters H, I, O, and X all have point and line symmetry.
Centre of Symmetry Examples For Alphabets
Point Symmetry In Shapes
If two equal forms are made by putting a point on an object or shape, but they face different directions, the object or shape shows point symmetry. Or in simple words, we can say that the point of origin should be equally distant from the two opposite sides of a shape. Squares and rectangles, for example, have a point of symmetry. We can restore the original form by rotating the shapes square and rectangle 180 degrees.
Conclusion
In conclusion, we can say that every object may or may not be symmetrical depending upon the line of symmetry that passes through it. An object has a centre of symmetry if an imaginary line can be stretched from any point on its surface to its centre and a corresponding point exists along the line equidistant from the centre.
Sample Questions
1. Which alphabet has a point of symmetry?
a. Q
b. O
c. G
d. L
Ans. O
Explanation: A point of symmetry is a point of origin through which a line of symmetry passes. O can be divided into two equal-looking halves making it an alphabet not having a line of symmetry with two parts being placed equally distant from each other.
2. Which shapes do not have a point of symmetry?
a. Equilateral Triangle
b. Circle
c. Square
d. None of the above
Ans. None of the above
Explanation: All the shapes have points of symmetry as they have two opposite parts that are equally placed from the point of origin.
3. A line of symmetry divides the object into two or more parts.
a. True
b. False
Ans. False
Explanation: A line of symmetry divides the object into two equal halves which when placed over one another cover themselves fully.
FAQs on Point Symmetry in Geometry Explained with Clear Examples
1. What is point symmetry in Maths?
Point symmetry is a type of symmetry where a figure looks the same after a 180° rotation about a fixed point called the centre of symmetry. In point symmetry, every point on the shape has a matching point directly opposite through the centre at the same distance. If you rotate the figure half a turn (180°), it appears unchanged. This is also known as rotational symmetry of order 2.
2. How do you know if a shape has point symmetry?
A shape has point symmetry if it looks identical after a 180° rotation about its centre. To check:
- Identify a possible centre point.
- Rotate the shape 180° around that point.
- If the rotated image matches the original exactly, the shape has point symmetry.
For example, a rectangle has point symmetry because it matches itself after a half-turn.
3. What is the centre of symmetry?
The centre of symmetry is the fixed point about which a figure can be rotated 180° and still look the same. In coordinate geometry, it is the midpoint between corresponding opposite points. For example, in a parallelogram, the intersection of the diagonals is the centre of symmetry.
4. Which shapes have point symmetry?
Shapes that have point symmetry include those that remain unchanged after a 180° rotation. Common examples are:
- Parallelogram
- Rectangle
- Rhombus
- Square
- Circle
A regular polygon has point symmetry if it has an even number of sides, such as a regular hexagon.
5. Does a circle have point symmetry?
Yes, a circle has point symmetry about its centre. In fact, a circle looks the same after rotation by any angle, including 180°. This means it has infinite rotational symmetry and therefore satisfies the condition for point symmetry.
6. What is the difference between line symmetry and point symmetry?
The main difference is that line symmetry involves reflection across a line, while point symmetry involves rotation of 180° about a point.
- Line symmetry: The shape mirrors across a line (axis of symmetry).
- Point symmetry: The shape matches after a half-turn around a centre.
A rectangle has both line symmetry and point symmetry, but a general parallelogram has only point symmetry.
7. How do you find point symmetry on a coordinate plane?
To find point symmetry on a coordinate plane, check whether corresponding points are equal distances from a common midpoint. Steps:
- Choose a suspected centre, such as the origin.
- For each point (x, y), check if (−x, −y) also exists.
- If yes, the figure has point symmetry about that centre.
For example, the graph of y = x³ has point symmetry about the origin (0,0).
8. Can a figure have point symmetry but no line symmetry?
Yes, a figure can have point symmetry without having line symmetry. A common example is a general parallelogram. It matches itself after a 180° rotation about the intersection of its diagonals, but it does not have any mirror line (unless it is a rectangle or rhombus).
9. What is an example of point symmetry with coordinates?
An example of point symmetry is the pair of points (2, 3) and (−2, −3), which are symmetric about the origin. The origin is the midpoint because:
- Midpoint x-coordinate = (2 + (−2)) / 2 = 0
- Midpoint y-coordinate = (3 + (−3)) / 2 = 0
Since both points are equal distances from (0,0) in opposite directions, they show point symmetry about the origin.
10. Is point symmetry the same as rotational symmetry?
Point symmetry is a special case of rotational symmetry where the rotation is exactly 180°. All shapes with point symmetry have rotational symmetry of order 2. However, rotational symmetry can occur at other angles (like 90° or 60°), which may not necessarily imply only point symmetry.
