NCERT Solutions Ganita Prakash Chapter 9 Punching Game, Paper Cutting and Lines of Symmetry — Figure it Out

Two holes made by one punch are always mirror images of each other in the fold line. So the fold is the line that is exactly halfway between them and at right angles to the line joining them.

• (a) The two holes are side by side at the same height. The paper was folded along the vertical line through the middle.
• (b) The two holes lie slanting, one above the other towards the top corner. The paper was folded along the diagonal of the square (from the bottom-left corner to the top-right corner).
• (c) The two holes are one above the other near the right edge. The paper was folded along the horizontal line through the middle.
• (d) There are four holes, one at each corner, from a single punch. So the paper was folded twice — first in half vertically and then in half horizontally (or the other way round). The punch then went through 4 layers.

For each figure, measure how far the given hole is from the mirror line, and mark the new hole at the same distance on the other side, on the line through the hole at right angles to the mirror.

• (a) Square with a diagonal mirror — the hole is near the left edge, close to the top-left corner. Its mirror lies just as far below the top edge as the given one is from the left edge, so the new hole sits near the top edge. (1 new hole.)
• (b) Rectangle with a horizontal mirror — the hole is below the line, near the right edge. The new hole goes directly above it, the same distance above the line. (1 new hole.)
• (c) Triangle with a vertical mirror — the hole is just left of the line, so the new hole is just right of the line, at the same height. (1 new hole.)
• (d) and (e) Circle with a slanting diameter — drop a perpendicular from the hole to the diameter, continue it the same distance beyond, and punch there. (1 new hole in each.)

This part only explains the two pictures that will be used in question 4. Learn to read them:

After one fold the paper is 2 layers thick, so a single cut produces 2 identical marks. After two folds it is 4 layers thick and one cut produces 4 marks.

In every case the answer is found the same way: take the piece cut away and add its mirror image in the crease.

• (a) The cut piece touches the fold and has a V-shaped dent on its far side. On opening you get a single hole in the middle of the sheet with two points at the top, two points at the bottom and a V-dent on each side — a butterfly-like hole. The crease is its line of symmetry.
• (b) A band with a swallow-tail (a “<” notch) is cut from the fold. Opening gives a long ribbon-shaped hole right across the middle, with a V-notch at each end. This hole has two lines of symmetry — the crease, and the horizontal line along the middle of the band.
• (c) Here the sheet is folded twice (once vertically, then once horizontally), so it is 4 layers thick. The two small square notches cut on the folded corner open out into four notches, arranged symmetrically about both creases; the notch that lay on a crease becomes a hole of double size. The opened sheet has both a vertical and a horizontal line of symmetry.
• (d) One vertical fold. The bracket cut on the folded edge opens into a rectangular hole in the middle of the sheet, and the bracket cut on the free edge opens into two notches, one on the left edge and one on the right edge.

Fold the sheet twice so that the centre of the sheet comes to one corner of the folded packet. That corner is the closed corner — the centre of the paper. One straight cut near it removes a piece from all 4 layers at once, and opens out as a hole in the middle.

(a) Upright square hole

1. Fold the sheet in half horizontally, then fold it in half vertically. The centre of the sheet is now the closed corner of the small square packet.
2. Cut a small square notch at that closed corner, with its two straight sides running along the two creases.
3. Open out — the four quarter-notches join to make one square hole in the centre.

(b) Square hole standing on a corner (a “diamond”)

1. Fold horizontally and then vertically, exactly as before.
2. Now make a single slanting straight cut across the closed corner, cutting off a small right-angled triangle with its two equal short sides on the creases.
3. Open out — the four triangles together form a square hole turned through 45°.

a. The first shape is a square standing on one corner — turning a square does not change it, so it still has 4 lines of symmetry: two through opposite corners and two through the midpoints of opposite sides.
The second shape is a regular eight-pointed star — it has 8 lines of symmetry: 4 through opposite points and 4 through opposite inner corners.

b. A triangle with equal sides and equal angles is an equilateral triangle. It has 3 lines of symmetry — one from each corner to the midpoint of the opposite side.

c. A hexagon with equal sides and equal angles is a regular hexagon. It has 6 lines of symmetry — 3 joining opposite corners and 3 joining the midpoints of opposite sides.

The four rhombus (diamond) patterns:

The four figures drawn on squared paper:

• Square with a smaller square inside, divided into four — 4 lines of symmetry (vertical, horizontal, both diagonals). The inner square sits exactly in the middle, so it does not spoil any of them.
• The eight-sided figure (octagon-like) — 2 lines of symmetry (vertical and horizontal). It is taller than it is wide, so the diagonals are not lines of symmetry.
• The five-sided figure — 1 line of symmetry, the horizontal line through the left corner.
• The four-pointed star — 4 lines of symmetry (vertical, horizontal and both diagonals).

The kolam is built from seven six-pointed stars — one in the middle and six around it — with small rhombuses filling the gaps. That makes it a hexagonal design.

The kolam has 6 lines of symmetry, all passing through the centre star:

• 3 lines that pass through pairs of opposite outer stars;
• 3 lines that pass between the stars, through the rhombuses.

• a. Draw an isosceles triangle — two sides equal, the third different (say 6 cm, 6 cm and 4 cm). Its only line of symmetry runs from the top corner to the midpoint of the unequal side.
• b. Draw an equilateral triangle — all three sides equal (say 5 cm each). It has 3 lines of symmetry.
• c. Draw a scalene triangle — all three sides different (say 4 cm, 6 cm and 7 cm). It has no line of symmetry.

Is a triangle with exactly two lines of symmetry possible? No, it is impossible.

• a. Exactly one line: a semicircle (half a chapati). Its only line of symmetry is the perpendicular to the straight edge through its midpoint. A “D” shape, a heart or an ice-cream cone with a rounded top will also do.
• b. Exactly two lines: an oval (ellipse), or a rectangle with the two short sides replaced by semicircular bumps. The lines of symmetry are the long axis and the short axis.
• c. Exactly four lines: a flower with four equal petals drawn round a centre, or a square with a semicircular bump on each side. The lines are the vertical, the horizontal and the two diagonals.

Work square by square on the grid. For every corner of the given red shape, count the number of squares from the blue line, go the same number of squares on the other side, and mark the matching corner. Then join your new corners in the same order.

• (a) Done in the book: the half-arrow on the right of the vertical blue line is copied to the left, giving a kite-shaped figure.
• (b) The blue line is horizontal. Copy the stepped shape downwards: every point that is 3 squares above the line goes 3 squares below it. The finished figure looks like the same “tower” standing on its own reflection.
• (c) The blue line is a slanting (diagonal) line. Turn your book so that this line becomes vertical — then the copying is easy again. On a diagonal mirror, a point that is a squares right and b squares up from a point of the line comes back as b squares right and a squares up.
• (d) Vertical blue line, small hook on the left — copy it to the right; the two hooks together make a shape like a bracket “]—[”.
• (e) Horizontal blue line at the top with a five-sided shape hanging below — copy it above the line to get a closed figure shaped like a diamond with two flat ends.
• (f) Slanting blue line again; rotate the book, then reflect the hexagon-like shape to the other side of the line.

Do the reflection twice:

1. Reflect the given red piece in the first blue line.
2. Now reflect both pieces (the given one and the new one) in the second blue line.

You will end up with 4 copies of the piece, one in each of the four regions made by the two lines.

• (a) The two blue lines are the diagonals. The short red segment near the bottom must be copied into all four quarters; the finished figure looks like a square standing on a corner with four equal marks.
• (b) Diagonals again, with a zigzag on the left. Reflect it in one diagonal, then reflect the pair in the other — a four-armed “windmill with mirrors”.
• (c), (d), (e), (f) Here the blue lines are the vertical and the horizontal line. Reflect the given piece across the vertical line, then reflect both across the horizontal line, giving 4 identical pieces placed symmetrically about the centre.

Many different answers are possible. The easiest method is:

1. Choose the line that you want as the axis — usually a vertical, a horizontal or a 45° line through the dots.
2. Reflect the given lines in it. Because two of the given corners usually fall on the axis, exactly two new lines are needed to close the shape.

Here is one working answer for each of the six figures (using columns numbered 0–5 from the left and rows 0–5 from the top):