ICSE Class 8 Maths Chapter 27 Symmetry Reflection

Chapter 27: Symmetry, Reflection

John Keats wrote 'A thing of beauty is joy forever'. When we describe an object as beautiful, perhaps we mean that it is balanced and has regularity in shape.

Line Symmetry

Look at the following plane figures:

We observe that if these figures are folded along a specific line (shown dotted), each figure on the left hand side of the dotted line fits exactly on the top of the figure on the right hand side of the dotted line i.e. each figure is divided into two coincident parts about the dotted line. This leads to:

If a figure is divided into two coincident parts by a line then the figure is called symmetrical about that line. The line which divides the figure into two coincident parts is called the line of symmetry or axis of symmetry or mirror line.

A simple test to determine whether a figure has line symmetry is to fold the figure along the supposed line of symmetry and to see if the two halves of the figure coincide.

Each one of the figures (i), (ii) and (iii) shown above is symmetrical about the dotted line, and the dotted line is its axis (or line) of symmetry.

A two dimensional (plane) figure may or may not have a line of symmetry. Moreover, a plane figure may have more than one line of symmetry.

Teacher's Note

When looking at butterfly wings or a person's face, we notice they have perfect line symmetry - one side mirrors the other exactly, which our brains find naturally beautiful and balanced.

Point Symmetry

Point symmetry exists when a figure is built around a single point called the centre of the figure. For every point of the figure, there is another point found directly opposite it at the same distance on the other side of the centre.

A simple test to determine whether a figure has point symmetry is to turn it upside-down and see if it looks the same. A figure that has point symmetry is unchanged in appearance by 180 degree rotation.

Parallelogram

Let ABCD be a parallelogram and O be the point of intersection of its diagonals. It has a point symmetry about the point O because every point P on the figure has a point P' directly opposite it on the other side of O.

Teacher's Note

Window panes in older buildings often have diamond or parallelogram patterns that exhibit point symmetry - if you rotate them 180 degrees, they look identical to what they were before.

Letter S

The letter 'S' has a point symmetry about the point O because every point P on the figure has a point P' directly opposite it on the other side of O.

Rotation

Look at the adjoining sketch. Let us rotate the line segment AB through 90° (clockwise) about the point O.

Steps

1. Join OA and OB.

2. Draw \(\angle AOA' = 90°\) and \(\angle BOB' = 90°\) (clockwise) such that \(OA' = OA\) and \(OB' = OB\).

3. Join A'B'.

We say that the line segment AB has been rotated through 90° (clockwise) about the point O. The point O is called centre of rotation.

Notice that \(A'B' = AB\) (measure and check it).

Thus, a figure can be rotated through any angle (clockwise or anticlockwise direction) about a point, and the point is called centre of rotation.

On rotation, the size of the figure remains the same.

Rotational Symmetry

Look at the adjoining sketch. It does not have any line of symmetry or point of symmetry. Yet it seems balanced and has regularity of shape. Let this figure be rotated through one complete turn (clockwise or anticlockwise) about the point O. There are three occasions when it looks the same as it did in its starting position. These are when it has been rotated through 120°, 240° and 360°. We say that this figure has a rotational symmetry of order 3.

Note that if A° is the smallest angle through which a figure can be rotated and still look the same, then it has a rotational symmetry of order = \[\frac{360}{A}\]

Remember that for a figure to have a rotational symmetry, A° must be less than or equal to 180°.

Symmetry of Some Figures

In each of the following case, the dotted line (or lines) is the axis of symmetry:

Line Segment

A line segment has

(i) one line of symmetry - the perpendicular bisector of the segment.

(ii) one point of symmetry - the mid-point of the segment.

(iii) rotational symmetry of order 2.

Angle

An angle (with equal arms) has

(i) one line of symmetry - the bisector of the angle.

(ii) no point of symmetry.

(iii) no rotational symmetry.

Scalene Triangle

A scalene triangle has

(i) no line of symmetry.

(ii) no point of symmetry.

(iii) no rotational symmetry.

Isosceles Triangle

An isosceles triangle has

(i) one line of symmetry - the bisector of the angle included between equal sides.

(ii) no point of symmetry.

(iii) no rotational symmetry.

Equilateral Triangle

An equilateral triangle has

(i) three lines of symmetry - the bisectors of the angles.

(ii) no point of symmetry.

(iii) rotational symmetry of order 3.

The centre of rotation is the point of intersection of the bisectors of the angles.

Parallelogram

A parallelogram has

(i) no line of symmetry.

(ii) one point of symmetry - the point of intersection of the diagonals.

(iii) rotational symmetry of order 2.

The centre of rotation is the point of intersection of the diagonals.

Teacher's Note

A kite shape or a butterfly exhibit symmetries in nature - the diagonal line through the center divides each shape into two perfectly matched halves that mirror each other.

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