9. Symmetry

Both have a vertical line of symmetry through the middle. The left half is a mirror image of the right half.

Flower: 6 lines of symmetry. Rangoli: 4 lines. Butterfly: 1 line.

The pinwheel has no line of symmetry (it has only rotational symmetry). The cloud has no line of symmetry.

1st figure (arrow-like shape): 1 line, through the notch and the opposite corner.

2nd figure (kite): 1 line, the vertical line through the top and bottom corners.

3rd figure (shape with a slanting top): no line of symmetry.

4th figure (L-shape): 1 line, the slanting line through the outer corner and the inner corner.

5th figure (triangle with unequal sides): no line of symmetry.

No other way. A square has 4 lines of symmetry: 1 vertical, 1 horizontal and 2 diagonals.

Snowflake-like figure: 6 lines. Yellow flower: 6 lines. Green knot design: 4 lines.

No. When we fold a rectangle along its diagonal, the two parts do not overlap exactly.

Along diagonal AC: A and C stay where they are. B and D change places.

Along the horizontal line: A and D change places, and B and C change places.

Yes, the figure is symmetric. The fold line is the line of symmetry. Usually there is no other line of symmetry.

The cut shape appears on both sides of the fold as mirror images. The fold line is the line of symmetry.

(a) Folded along the vertical line through the middle.

(b) Folded along the diagonal joining the top-right and bottom-left corners.

(c) Folded along the horizontal line through the middle.

(d) The paper was folded twice — once vertically and once horizontally — and one hole was punched near a corner.

(a) The other hole is near the top side, at the same distance from the top-left corner (the reflection across the diagonal).

(b) The other hole is near the top-right corner, as far above the middle line as the given hole is below it.

(c) The other hole is on the right of the vertical line, at the same height.

(d) The other hole is just below the slanting diameter, opposite the given hole.

(e) The other hole is just below the slanting line, near the left part of the circle, opposite the given hole.

In each case, the cut shape is repeated as a mirror image on the other side of every fold.

(a) and (b) have one fold, so the hole is a shape made of two mirror halves.

(c) and (d) are folded more than once, so the cut appears on all the folded layers, giving symmetric notches as shown.

(b) Tilted square (diamond) hole: Fold the paper in half and then in half again. The centre of the paper is now at the folded corner. Cut straight across this corner.

(a) Upright square hole: Fold the paper in half and in half again, and then fold once more along the diagonal through the centre corner. Now make one straight cut across, perpendicular to the folded edge.

(a) Square: 4 lines. 8-pointed star: 8 lines.

(b) Equilateral triangle: 3 lines.

(c) Regular hexagon: 6 lines.

The kolam has 2 lines of symmetry — one vertical and one horizontal, through the centre star.

The slanting parallelograms stop the slanting lines from being lines of symmetry.

(a) An isosceles triangle (two equal sides), for example sides 5 cm, 5 cm and 3 cm.

(b) An equilateral triangle (all sides equal).

(c) A scalene triangle (all sides different), for example 3 cm, 4 cm and 6 cm.

No. If a triangle has two lines of symmetry, all its sides become equal, and then it has three lines.

(a) 1 line: a semicircle, or a heart shape.

(b) 2 lines: an oval (ellipse), or the shape of an eye.

(c) 4 lines: a square with a semicircle on each side, or a circle inside a square.

For each corner of the red figure, count the squares to the blue line. Mark a point the same number of squares on the other side. Join the points in the same order.

For (c) and (f), the blue line is slanting. Turn the book so that the line looks straight, and then reflect.

First reflect the red part across one blue line. Then reflect the whole drawing (old part and new part) across the other blue line. The final figure has four matching parts.

Find a line that cuts the drawn part into two mirror halves, or a line about which one drawn side can be reflected. Draw the two missing sides as the mirror images of the existing sides so that the figure closes.

A plus sign (+) with equal arms has exactly four angles of symmetry.

A square has 4 angles of symmetry: 90°, 180°, 270° and 360°. The centre of rotation is the point where the diagonals meet.

It has 4 angles of symmetry: 90°, 180°, 270° and 360°.

If we change the angles between the arms, it no longer has 4 angles of symmetry. The arms must be 90° apart.

For only 2 angles of symmetry (180° and 360°), make the arms meet at angles like 60°, 120°, 60°, 120°, or add flaps that turn the same way on opposite arms.

5 arms, 72° apart (360° ÷ 5): angles of symmetry 72°, 144°, 216°, 288°, 360°.

6 arms, 60° apart (360° ÷ 6): angles of symmetry 60°, 120°, 180°, 240°, 300°, 360°.

360° ÷ 7 = 5137°. It is not a whole number.

(a) Plus shape: 90°, 180°, 270°, 360°

(b) Bracket-like shape: only 360° (no rotational symmetry)

(c) H-shape: 180°, 360°

All except the last figure (made of two half circles) have more than one angle of symmetry:

circle with two diameters (180°, 360°), triangle with equal sides (120°, 240°, 360°), circle divided into 3 equal parts (120°, 240°, 360°), four-bladed pinwheel (90°, 180°, 270°, 360°), the cross X (180°, 360°) and the five-pointed star (72°, 144°, 216°, 288°, 360°).

Line with arrowheads at both ends: 2  |  X with one marked angle: 1  |  Six-pointed star: 6

Three-armed figure: 3  |  Plus shape: 4  |  Regular pentagon: 5

Yes. If a figure looks the same after turning through the smallest angle, it looks the same again after turning through it twice, three times, and so on.

(1) True. A full turn brings every figure back to its starting position.

(2) True. The multiples of the smallest angle must reach exactly 360°.

(i) Colour every 4th sector (sectors 1, 5 and 9). Angles: 120°, 240°, 360°.

(ii) Colour every 3rd sector (sectors 1, 4, 7 and 10). Angles: 90°, 180°, 270°, 360°.

(iii) The possible numbers are 1, 2, 3, 4, 6 and 12 (the factors of 12).

An equilateral triangle (3 lines, angles 120°, 240°, 360°).

A regular hexagon (6 lines, 6 angles). A rectangle and a plus sign also work.

(a) An equilateral triangle (3 lines, 3 angles of symmetry).

(b) An isosceles triangle that is not equilateral.

(c) A parallelogram that is not a rectangle (angles of symmetry 180° and 360°).

(d) A kite, or an isosceles trapezium.

120°, 180°, 240°, 300° and 360°.

The angles less than 60° must be 20° and 40°. So the smallest angle is 20°.

(a) Yes. 45° × 8 = 360°. A figure with 8 equal radial arms is an example.

(b) No. 17 is not a factor of 360.

(a) Yes. It has 3 lines of symmetry.

(b) Yes. The angles of symmetry are 120°, 240° and 360°.

Triangle 3, square 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.

We get the counting numbers starting from 3. A regular polygon has as many lines of symmetry as sides.

Also 3, 4, 5, 6, 7, 8, 9, 10 — the counting numbers starting from 3.

The first shape (triangle) has 3 lines and 3 angles of symmetry.

Every later shape has 6 lines and 6 angles of symmetry.

The Ashoka Chakra has 24 spokes. It has 24 lines of symmetry and 24 angles of symmetry (multiples of 15°).

The red lines in the figure are the two lines of symmetry. Copy the coloured top-left quarter into the other three quarters as mirror images across the red lines.

With 16 tiles, arrange a 4 × 4 square and colour it so that one half mirrors the other across one line (1 line), or across both middle lines (2 lines).

The second player can always win. After each move of the first player, the second player covers the two squares that are the exact opposite (turned by 180° about the centre of the grid).

Such a move is always free, so the first player is the one who runs out of moves.