Line of Symmetry: A line of symmetry, also known as an axis of symmetry or mirror line, is a line that splits a shape or object into two identical and symmetrical halves. Each half mirrors the other, making them look like reflections of one another across the line which indicates that both parts of the shape would perfectly match. The shape would not change if a mirror were put along this line.
In this article, we will talk about the concept of line of symmetry, along with its definition, and properties. We will also explore lines of symmetry in alphabets, and lines of symmetry in shapes like Triangle, Rhombus, Parallelogram, Pentagon, etc.
What is Line of Symmetry?
A Line of symmetry is an imaginary line that divides or separates a shape or an object into symmetrical halves. In simple words, one half is the reflection of the other half.
In Taj mahal, we observe that one-half on one side of Taj mahal is the same as the one-half on the other side because each side is a perfect reflection or mirror image of the other side.
If you cut an object along its axis, it will create a mirror-like reflection of either side of the object. This axis is known as the axis of symmetry. For example, if we cut starfish and octopus along their axis of symmetry, we will get similar shapes.
Line of Symmetry Meaning
A line of symmetry is a line that divides an object into two equal parts. These two parts are the mirror reflection of each other. A line of Symmetry is an imaginary line that passes through the center of that object.
Properties of Line of Symmetry
There are a few properties to understand the concept of line of symmetry effectively:
- If a body does not contain any line of symmetry, then that body implies asymmetrical.
- A shape or body can have Infinite symmetry lines. For example, in circles.
- An object can have one symmetry line. For example, a butterfly's wings have only one line of symmetry along its vertical axis.
- Some objects can have two symmetry lines.
Check: Symmetry
Line of Symmetry Types
Generally, three types of lines of symmetry can be observed in different geometrical figures and they are:
- Horizontal line of symmetry
- Vertical line of symmetry
- Diagonal line of symmetry
Horizontal Line of Symmetry
Horizontal Symmetry Line is a sleeping straight line that divides a geometrical shape into two equal halves in a horizontal way. It goes from left to right or right to left in an object.
Vertical line of Symmetry
Vertical Symmetry Line is a standing straight line that divides a geometrical shape into two equal halves in a vertical way. It goes from top to bottom or vice versa in an object.
Diagonal line of Symmetry
Diagonal Symmetry Line is a slanted line that divides a geometrical shape into two equal halves in a sideway.
Line of Symmetry for Different Shapes
In geometry, we have shapes with one or multiple line of symmetry as per its structure such as the square, rectangle, triangle, rhombus and parallelogram. These lines can be horizontal, vertical or diagonal. In two dimensional shapes we will learn the lines of symmetry for following shapes
- Line of Symmetry in Triangle
- Line of Symmetry in Quadrilateral
- Line of Symmetry in other Polygons
- Line of Symmetry in other Shapes
Line of Symmetry in Triangle
Triangle is a polygon bounded by three-line segments. Based on the length of sides, there are three types of triangles namely Equilateral Triangle, Isosceles Triangle and Scalene Triangle.
Line of Symmetry in Equilateral Triangle(Three Line Symmetry)
Equilateral Triangle is a triangle in which all sides are equal. Hence, an Equilateral Triangle has three lines of Symmetry. The symmetry lines pass from the vertex and the midpoint of opposite side.
Line of Symmtery in Isosceles Triangle(One Line Symmetry)
Isosceles Triangle is a triangle in which only two sides are equal. Hence, an Isosceles Triangle has only one lines of symmetry.
Line of Symmetry in Scalene Triangle(No Symmtery Lines)
A scalene triangle is a triangle in which no sides are equal. Hence, Scalene Triangle has no lines of symmetry.
Line of Symmetry in Quadrilateral
Quadrilateral is a polygon which has four sides. The different types of quadrilateral are trapezium, parallelogram, rhombus, square, rectangle, and kite. Let's learn about the symmetry lines in different Quadrilateral.
Line of Symmetry in Trapezium (No Symmetry Lines)
A trapezium is a quadrilateral in which a pair of opposite sides are parallel. A Trapezium has no symmetry lines.
Line of Symmetry in Parallelogram(No Symmetry Lines)
A Parallelogram is a quadrilateral in which opposite sides are parallel and equal. A parllelogramn has no lines of symmetry.
Line of Symmetry in Kite (One Line Symmetry)
A kite contains one symmetry line. This symmetry line in kite is vertical in nature.
Line of Symmetry in Rectangle (Two Line Symmetry)
A rectangle has two symmetry lines: one vertical and one horizontal symmetry line. These lines pass through the midpoints of opposing sides. When it is folded diagonally, it gives asymmetrical shape.
Line of Symmetry in Rhombus (Two Line Symmetry)
A rhombus has two symmetry lines. These two lines of symmetry in rhombus are its diagonals.
Line of Symmetry in Square (Four Line Symmetry)
A square has four lines of symmetry: one vertical, one horizontal, and two diagonal symmetry lines. The four lines of symmetry are formed by lines through the midpoints of opposite sides.
Line of Symmetry in Higher Polygons
We know that a polygon is a closed figure bounded by three or more-line segments. Above we have learnt about the lines of symmetry in triangle and quadrilaterals. Let's learn about lines of symmetry in higher polygons such as pentagon, hexagon etc.
Line of Symmetry in Pentagon (Five Line Symmetry)
A regular pentagon has five lines of symmetry: one horizontal, one vertical, and three diagonal.
Line of Symmetry in Hexagon (Six Line Symmetry)
A regular hexagon contains six lines of symmetry: one horizontal, one vertical, and four diagonal.
Line of Symmetry in Heptagon(Seven Line Symmetry)
A regular Heptagon has 7 equal sides. Hence, there are 7 lines of symmetry in Heptagon.
Line of Symmetry in Octagon (Eight Line Symmetry)
An octagon contains eight symmetry lines. Out of these 8 symmetry lines four are horizontal lines of symmetry and four are vertical symmetry lines
Line of Symmetry in Other Shapes
In this heading we learn the lines of symmetry for other shapes such as Circle and Star. Since a circle can fold along the lines of diameter, hence all the lines of symmetry of a circle will be along its diameter. In case of star, the lines of symmetry are along the lines joining the opposite vertices.
Line of Symmetry in Circle (Infinite Symmetry Lines)
A circle has an infinite number of symmetry lines because it is symmetrical along its infinite axis and endless number of lines can be passes through its centre.
Line of Symmetry in Star (Five Line Symmetry)
In Star, the lines of Symmetry pass through the opposite vertices. Hence, in a star there are total five lines of symmetry.
Line of Symmetry in Alphabets
Apart from geometrical shapes, English alphabets also have lines of symmetry. Below mentioned topics explain the lines of symmetry in English alphabets.
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Vertical Line of Symmetry in English Alphabets
English alphabets with respect to vertical line of symmetry are A, H, I , M, O, T, U, V, W, X, Y.
Horizontal Line of Symmetry in English Alphabets
English alphabets with respect to horizontal line of symmetry are B, C, D, E, H, I, K, O, X.
Two Line of Symmetry (Horizontal and Vertical) in English Alphabets
The letters which have horizontal and vertical lines of symmetry are H, I , O , X.
Zero Line of Symmetry in English Alphabets
Alphabets such as F, G, L, P, Q, R, S, Z have no line of symmetry.
Line of Symmetry Equations
There are several ways to find the line of symmetry. In the coordinate system, the graph may have symmetry about the x-axis, y-axis, origin, or about any given line.
For Example, a parabola has line symmetry in coordinate geometry and line symmetry equation for a parabola with the quadratic equation y = ax2 + bx + c is of the form x = n, where n is a real number and a and b are the coefficients of x2 and x respectively and c is the constant. x = -b/(2a) gives us our line of symmetry.
This means that the axis of symmetry is the vertical line passing through x = -b/(2a). This line will split the parabola into two equal halves.
For example, if we are given the equation for a parabola in standard form such as y = x2 + 6x + 3, we know that a=1, b=6, and c=3. Here, we can put the value of a and b into the formula x=−b/2a in order to get the axis of symmetry. The formula becomes x =−(6)/2(1) which results to x = −3. This means that the axis of symmetry is the vertical line passing through x=−3.
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Line of Symmetry Examples
Example 1. Find the axis of symmetry of the parabola y = -2x2 − 4x + 8.
Solution:
Given equation: y = -2x2 − 4x + 8
Compare the given equation with the standard form ax2 + bx + c.
⇒ a = -2, b = −4, c = 8
Use the formula of axis of symmetry = −b/2a
= −(−4)/2(-2) ⇒ x = 1
The equation of the line for the axis of symmetry is x=1.
Example 2. Find the axis of symmetry of the parabola y = 4x2.
Solution:
Given equation : y = 4x2
Compare the given equation to the standard form ax2 + bx + c.
⇒ a = 4, b = 0, c = 0
Use the formula of axis of symmetry = −b/2a
= -0/2(4) ⇒ x = 0
Example 3. How do you locate a shape’s Lines of Symmetry?
Solution:
Firstly, we will fold the shape along a line in such a way that one half is exactly similar to the other half. That line is known as the line of symmetry.
Example 4. How many symmetry lines can be drawn on a regular pentagon?
Solution:
A line of symmetry splites a figure into two identical mirror images. Total 5 symmetry line can be drawn on regular pentagon: one horizontal, one vertical, and three diagonal.
Example 5. What is the standard formula for the Axis of Symmetry for a parabola?
Solution:
The standard formula of axis of symmetry of an equation (y=ax2 + bx + c) is x = −b/2a
Practice Questions on Line of Symmetry
Q1. Draw all the lines of symmetry for the equilateral triangle.
Q2. Draw all the lines of symmetry for the regular pentagon.
Q3. How many line of symmetry can be drawn in circle?
Q4. How many line of symmetry can be drawn in isosceles triangle and scalene triangle?
Q5. How many alphabets have exactly two lines of symmetry?
Q6. How many alphabets have exactly one lines of symmetry?
Q7. Write the line of symmetry for the quadratic equation y = x2 +18x+8.
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