| Classwise Concept with Examples | ||||||
|---|---|---|---|---|---|---|
| 6th | 7th | 8th | 9th | 10th | 11th | 12th |
| Content On This Page | ||
|---|---|---|
| Symmetry & Related Terms | Reflection and Symmetry | |
Chapter 13 Symmetry (Concepts)
Welcome to Chapter 13: Symmetry! Have you ever noticed how a butterfly's wings look the same on both sides? Or how folding a perfectly cut paper heart down the middle makes the two halves match exactly? This sense of balance and perfect matching is what we call symmetry in mathematics and art. It's a concept found everywhere in nature, design, and geometry. This chapter is all about understanding this beautiful idea, learning how to spot symmetry in different shapes, and identifying special lines called "lines of symmetry." Exploring symmetry helps develop our ability to visualize shapes and appreciate the patterns and harmony around us.
The main type of symmetry we'll focus on is called reflection symmetry or line symmetry. Imagine you have a flat shape drawn on paper. If you can find a line along which you can fold the paper so that the two halves of the shape lie exactly on top of each other (they coincide perfectly), then the shape has line symmetry. That special fold line is called the line of symmetry or the axis of symmetry. Think of the line of symmetry as a mirror placed along the fold. One half of the shape is the perfect mirror reflection of the other half across that line.
We will look at many different shapes and familiar objects to see if they have lines of symmetry and, if so, how many they have. For example:
- A rectangle has two lines of symmetry (one vertical, one horizontal, passing through the center).
- A square has four lines of symmetry (one vertical, one horizontal, and two diagonal).
- An isosceles triangle (with two equal sides) has one line of symmetry, running from the top vertex down to the middle of the base.
- An equilateral triangle (with all three sides equal) has three lines of symmetry, each running from a vertex to the midpoint of the opposite side.
- A circle has infinitely many lines of symmetry – any line passing through its center is a line of symmetry!
- Some shapes, like a scalene triangle (all sides different), have no lines of symmetry at all.
A fun and important activity related to symmetry is completing a figure. Imagine you are given only half of a shape and the line of symmetry. Your task is to draw the other half so that the completed figure is perfectly symmetrical. This requires visualizing the mirror image across the line of symmetry and drawing it accurately. We'll practice this using grids or dotted lines to help guide our drawing.
This chapter relies heavily on looking at pictures, diagrams, and maybe even using mirrors or folding paper! It's a very visual topic. By identifying lines of symmetry in various shapes and learning to complete symmetrical figures, you will develop your geometric intuition, your ability to see patterns, and your understanding of balance and harmony in shapes – key skills for understanding geometry and appreciating the world around you.
Symmetry & Related Terms
Symmetry is a fascinating concept found everywhere in the world around us, from the smallest snowflake to the largest galaxies. We see it in the natural patterns of leaves, flowers, and animals, in the designs of buildings and art, and even in musical forms. In mathematics and geometry, symmetry is a precise property that describes how a shape or object looks the same after being moved, rotated, or flipped.
In this section, we will focus on a common type of symmetry called line symmetry, also known as reflectional symmetry.
What is Symmetry? (Line Symmetry)
A figure is said to possess Line Symmetry (or reflectional symmetry) if it can be folded along a straight line such that one half of the figure exactly matches or coincides with the other half. When folded along this line, the two parts are mirror images of each other.
Imagine holding a mirror along the line of fold; the reflection of one half in the mirror would look exactly like the other half of the figure.
A classic example from nature is a butterfly. If you imagine a vertical line running down the centre of its body, the pattern on its left wing is a mirror image of the pattern on its right wing. When folded along this central line, the wings would overlap perfectly.
Many geometrical shapes, letters of the alphabet, and everyday objects exhibit this kind of balance and identical correspondence between two halves.
Line of Symmetry (or Axis of Symmetry)
The straight line that divides a figure into two identical halves that are mirror images of each other is called the Line of Symmetry or the Axis of Symmetry. This is the line along which you would fold the figure for the halves to match perfectly.
A figure can have one, multiple, or even infinitely many lines of symmetry, or it might have no line of symmetry at all.
Figures with One Line of Symmetry
Many figures are symmetrical, but only about one specific line. This means there is only one "fold line" that will cause the two halves of the figure to perfectly overlap. Such a line is called a line of symmetry or an axis of symmetry.
Isosceles Triangle: A triangle with two equal sides has exactly one line of symmetry. This line passes through the vertex formed by the two equal sides and bisects the opposite side (the base) at a right angle. This single line acts as an altitude, a median, and an angle bisector.
Kite: A kite, which has two distinct pairs of equal-length adjacent sides, has one line of symmetry. This line is the diagonal that runs between the vertices where the unequal sides meet.
Semicircle: A semicircle has one line of symmetry. This line is the perpendicular bisector of the diameter, passing through the center of the original circle.
Capital Letters: Several letters in the English alphabet have exactly one line of symmetry. These can be categorized as having either vertical or horizontal symmetry.
- Vertical Symmetry: A, M, T, U, V, W, Y
- Horizontal Symmetry: B, C, D, E, K
Figures with Multiple Lines of Symmetry
Some geometric figures possess more than one line of symmetry. They can be folded along several different lines to produce matching halves.
Rectangle: A rectangle has two lines of symmetry. These lines connect the midpoints of its opposite sides. Note that the diagonals of a rectangle are not lines of symmetry.
Rhombus: A rhombus has two lines of symmetry, which are its two diagonals.
Square: A square is highly symmetrical and has four lines of symmetry. Two lines connect the midpoints of opposite sides, and the other two are its diagonals.
Regular Polygons: A regular n-sided polygon has 'n' lines of symmetry.
- An Equilateral Triangle (3 sides) has 3 lines of symmetry. Each line connects a vertex to the midpoint of the opposite side.
- A Regular Hexagon (6 sides) has 6 lines of symmetry. Three lines connect opposite vertices, and three lines connect the midpoints of opposite sides.
Circle: A circle has infinitely many lines of symmetry. Any line that passes through the center of the circle (i.e., any diameter) is a line of symmetry.
Capital Letters: Some letters have two lines of symmetry.
- H, I, X: Have both a horizontal and a vertical line of symmetry.
- O: Has multiple (theoretically infinite if perfectly circular) lines of symmetry.
Figures with No Line of Symmetry
Figures that are not symmetrical are called asymmetrical. They have no line of symmetry, meaning there is no line along which they can be folded to create two matching halves.
Scalene Triangle: A triangle where all three sides have different lengths has no line of symmetry.
Parallelogram: A general parallelogram (that is not a rectangle or a rhombus) has no line of symmetry. Although its diagonals bisect each other, folding along either diagonal does not result in the two halves overlapping perfectly.
Irregular Quadrilateral: Any general quadrilateral with unequal sides and angles will have no line of symmetry.
Capital Letters: Many letters in the alphabet are asymmetrical.
- F, G, J, L, N, P, Q, R, S, Z
Example 1. Draw the lines of symmetry for the letter 'H'.
Answer:
We need to find the straight lines along which the letter 'H' can be folded so that the two halves match exactly.
- If we draw a vertical line right down the middle of 'H', the left half is the mirror image of the right half. Folding along this line would make the two halves coincide. This is one line of symmetry.
- If we draw a horizontal line right across the middle of 'H', the top half is the mirror image of the bottom half. Folding along this line would also make the two halves coincide. This is another line of symmetry.
The letter 'H' has two lines of symmetry: one vertical and one horizontal.
Reflection and Symmetry
In the previous section, we were introduced to the idea of symmetry, particularly line symmetry, where a figure can be folded along a line to create two matching halves. This concept of line symmetry is very closely related to the idea of reflection, like what you see when you look in a mirror.
Reflection (Mirror Image)
When you stand in front of a mirror, you see an image of yourself. This image is called a Reflection or a Mirror Image. The surface of the mirror acts as the 'mirror line'. Reflection is a transformation where every point in the original object has a corresponding point in the image, located on the opposite side of the mirror line but at the same distance from it.
Let's look at the key properties of a mirror image:
- Same Size: The reflected image is the same size as the original object. It is neither magnified nor reduced.
- Same Shape: The reflected image has the same shape as the original object.
- Lateral Inversion: The image is flipped horizontally (left and right are reversed). If you raise your left hand, your reflection appears to raise its right hand.
- Equal Distance from the Mirror Line: For any point on the original object, the distance from that point to the mirror line is exactly equal to the distance from its corresponding point in the mirror image to the mirror line.
- Perpendicularity: The straight line segment connecting any point on the original object to its corresponding point on the mirror image is perpendicular (forms a $90^\circ$ angle) to the mirror line.
Symmetry as Reflection
Now we can connect the concept of reflection back to symmetry. A figure possesses line symmetry if and only if it is its own mirror image when the mirror is placed along the line of symmetry.
In a figure with line symmetry, the line of symmetry serves as the mirror line. One half of the figure is the 'object', and the other half is its exact 'mirror image' across that line. When you fold the figure along the line of symmetry, the two halves overlap perfectly because one is a reflection of the other.
Example: Consider a square and its vertical line of symmetry passing through the midpoints of the horizontal sides.
If you place a mirror along the vertical line, the reflection of the left half of the square in the mirror will look exactly like the right half of the square. Similarly, the reflection of the right half will look exactly like the left half. This property confirms that the vertical line is a line of symmetry for the square.
Drawing Symmetrical Figures by Reflection
Understanding the relationship between symmetry and reflection allows us to complete a symmetrical figure if we are given only one half of it and the position of the line of symmetry. We simply need to draw the reflection of the given half across the line of symmetry to complete the whole figure.
Steps to Complete a Symmetrical Figure Using Reflection:
- Identify the given half of the figure and the line of symmetry.
- Choose several key points on the boundary of the given half (especially vertices or turning points).
- For each chosen point, measure its perpendicular distance from the line of symmetry.
- Starting from the point where the perpendicular line meets the line of symmetry, extend the perpendicular line to the opposite side of the line of symmetry, measuring the exact same distance as in Step 3. Mark this new point; it is the reflection of the original point.
- Repeat Step 3 and 4 for all the key points you identified.
- Connect the reflected points on the other side of the line of symmetry in the same way the original key points were connected in the given half. Draw any curved parts as mirror images of the original curves.
- The complete figure will be the original half combined with the newly drawn reflected half.
Example 1. Complete the symmetrical figure given half of it and the line of symmetry.
Answer:
We are given the left half of a symmetrical figure and the line of symmetry (the vertical line). To complete the figure, we need to draw the reflection (mirror image) of the given left half across the vertical line of symmetry.
We can take key points on the boundary of the given half and reflect them. Let's consider the vertices of the given left half, based on the example shape described in the note:
- The point at (0,0) is on the line of symmetry. Its reflection is itself (0,0).
- The point at (2,0) is 2 units to the right of the line of symmetry (x=0). Its reflection will be 2 units to the left of the line of symmetry, at (-2,0).
- The point at (2,3) is 2 units to the right of the line of symmetry. Its reflection will be 2 units to the left, at (-2,3).
- The point at (1,4) is 1 unit to the right of the line of symmetry. Its reflection will be 1 unit to the left, at (-1,4).
- The point at (0,3) is on the line of symmetry. Its reflection is itself (0,3).
Now, we connect these reflected points in the same order as the original points were connected in the given half.
- Connect (-2,0) to (-2,3).
- Connect (-2,3) to (-1,4).
- Connect (-1,4) to (0,3).
- Connect (0,3) to (0,0). (These last two segments complete the connection back to the original half).
The completed symmetrical figure is shown below. It is formed by the original left half and the reflected right half, and it is symmetrical about the vertical line.
