ICSE Class 6 Maths Chapter 22 Linear Symmetry

Chapter 22: Linear Symmetry

(Including Constructions On Symmetry)

22.1 Concept Of Symmetry (Linear Symmetry)

Consider a plane mirror mm'. If an object F is kept at a distance 'd' in front of the mirror, the image of the object, F' as seen in the mirror, is formed at distance 'd' behind the mirror.

Now, if Figure 1, given alongside, is folded about the mirror mm', the object F and its image F' coincide. Since the two parts of the figure coincide when the figure is folded along the mirror line mm', we say that the figure is symmetrical about the mirror line mm'. For this reason, the mirror line mm' is called the line of symmetry of the whole figure, i.e. the object F, its image F' and the mirror mm' taken as a whole.

Making The Concept More Clear

Fold a rectangular piece of paper as shown in Figure (a) below. Then cut a piece of any design from the folded side of the paper as shown in Figure (b) below.

When we unfold the paper, there emerges a design, as shown in Figure (c) above. It is clear that the pattern of the design is identical on both the sides of the folding line (crease) of the paper, as shown here by the dotted line AB. So, AB is the line of symmetry here.

A figure that is identical on both the sides of a line in it is said to be symmetrical about that line, and the line about which the figure is symmetrical is called the line of symmetry or the axis of symmetry.

In order to find whether or not a given figure is symmetrical about a line in it, fold the figure about that line. If the part of the figure that lies on one side of the line coincides with the part of the figure on the other side of the line, the figure is symmetrical about that line.

Examples

1. A line segment is symmetrical about its perpendicular bisector.

2. An angle (with equal arms) is symmetrical about its bisector.

3. An isosceles triangle has one line of symmetry. The line of symmetry is the bisector of the angle contained by the two equal sides.

4. An equilateral triangle has three lines of symmetry. Each of the three bisectors of angles is a line of symmetry.

A scalene triangle has no line of symmetry.

A figure may have many lines of symmetry, e.g. a circle.

5. A kite-shaped figure has one line of symmetry.

6. A figure of the shape of an arrow-head has one line of symmetry.

7. The letter 'A' has one line of symmetry.

8. The letter 'H' has two lines of symmetry.

9. A circle has an infinite number of lines of symmetry. Every line that passes through the centre of a circle is a line of symmetry.

22.2 Symmetric Point

Consider a point P and a line AB. From point P, draw PO perpendicular to AB and then extend PO up to point Q such that OP = OQ. Now fold the figure obtained about the line AB. What do you observe?

Points P and Q coincide.

In this case, point Q is said to be the symmetric point of the given point P with respect to the given line AB.

Infact, points P and Q are symmetric to each other with respect to the line AB. Also, line AB is the perpendicular bisector of the line segment PQ. Line AB is the line of symmetry for the whole figure obtained.

22.3 To Locate A Point That Is Symmetric To A Given Point With Respect To A Given Line

Given a point P and a line AB, we want to find the point that is symmetric to the given point P with respect to the given line AB.

For this, first draw PO perpendicular to AB, and then extend PO and cut OQ = OP.

Clearly, point Q is symmetric to the given point P with respect to the given line AB.

Thus, AB is the line of symmetry.

To check your construction, fold the figure about the line of symmetry AB; you will find that P and Q coincide, i.e. they occur at the same point.

22.4 Constructing The Line Of Symmetry When Two Fixed Points Are Symmetric With Respect To The Required Line

Given two fixed points P and Q, we want to construct a line so that P and Q are symmetric with respect to this line.

For this:

1. Join P and Q.

2. Draw perpendicular bisector of the line segment PQ.

The obtained perpendicular bisector AB is the required line of symmetry, i.e. with respect to AB, the two points P and Q are symmetric.

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