Class 6 Ganita Prakash Chapter 9 Symmetry Solutions

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Class 6 Ganita Prakash · Chapter 9
Symmetry

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🗝️ Key ideas of this chapter

• Line of symmetry: folding along it makes the two halves overlap exactly (mirror halves / reflection symmetry).
• Square 4 lines, rectangle 2, equilateral triangle 3, regular n-gon n lines, circle — every diameter.
• Rotational symmetry: the figure looks the same after a turn of less than 360° about the centre of rotation.
• Angles of symmetry are multiples of the smallest one; the smallest (whole number) is a factor of 360.
• Some figures have only lines (kite), some only turns (windmill), some both (square).

📖 Textbook Page 217

📖 Page 217Repeating parts

Look at the flower, butterfly, rangoli and pinwheel. What repeats in the rangoli?

1Flower: the same petal repeats all round the centre.

2Butterfly: the left wing is a mirror copy of the right wing.

3Rangoli: the red petals come back onto themselves after a 90° turn about the centre — four copies of the same part.

✅ Answer: Parts repeat in a definite pattern (by folding/mirroring or by turning).

📖 Textbook Page 218

📖 Page 218The pinwheel

In the pinwheel, what figure repeats along each side of the hexagon?

1The centre is a regular hexagon made of 6 triangles.

2On each side of the hexagon an equilateral triangle is attached.

3Going round the hexagon, the attached triangles repeat after every 60° turn.

✅ Answer: Triangles stuck on each side of the hexagon repeat as you go round (turning by 60°).

📖 Page 218Clouds

Why do we say the cloud picture is not symmetrical?

1No part repeats in a definite pattern; folding or turning never gives the same picture.

✅ Answer: There is no repeating pattern.

📖 Page 218Taj Mahal and Gopuram

What symmetries do you see in the Taj Mahal and the Gopuram?

1Taj Mahal: the left half is a mirror image of the right half (minarets, domes, arches, pool).

2Gopuram: the left and right halves match; storeys repeat as we go up.

✅ Answer: Both have a vertical line of symmetry; the gopuram also has repeating storeys.

📖 Textbook Page 219

📖 Page 219Figure it Out · Q1

Do you see any line of symmetry in the figures at the start of the chapter? In the cloud?

1Flower: a line through each petal and through the gap between petals.

2Butterfly: one vertical line through the body.

3Rangoli: 4 lines (vertical, horizontal and two diagonals).

4Pinwheel: no line of symmetry (it has turning symmetry).

5Cloud: no line of symmetry.

✅ Answer: Flower many, butterfly 1, rangoli 4, pinwheel none, cloud none.

📖 Page 219Figure it Out · Q2

Identify the line(s) of symmetry of the five figures, if any.

Find the line(s) of symmetry.

1Fig 1 (arrow-like hexagon): one line through the notch and the opposite corner.

2Fig 2 (kite): one line through the top and bottom corners.

3Fig 3 (trapezium with one slanting side): none.

4Fig 4 (L-shape with unequal arms): none.

5Fig 5 (scalene triangle): none.

Lines of symmetry (blue dashed).

✅ Answer: 1 line, 1 line, none, none, none.

📖 Textbook Page 220

📖 Page 220Folding a square

Does a square have only one line of symmetry? Fold and find all.

1Fold vertically, horizontally and along both diagonals — each time the halves overlap.

2No other fold works.

4 lines of symmetry

✅ Answer: 4 lines of symmetry.

📖 Textbook Page 221

📖 Page 221More lines

Find all the lines of symmetry of the three figures (snowflake-like star, six-petal flower, knot design).

1Star (6 points): 6 lines.

2Flower with 6 identical petals: 6 lines (through petals and between them).

3Knot design: 4 lines (vertical, horizontal, two diagonals).

✅ Answer: 6, 6 and 4 lines.

📖 Page 221Diagonal of a rectangle

Is the diagonal of a rectangle (not a square) a line of symmetry?

1Fold along the diagonal: the two triangles do not overlap — the corners stick out.

2A rectangle has only 2 lines: through the midpoints of opposite sides.

✅ Answer: No. A rectangle has 2 lines of symmetry, not along the diagonals.

📖 Textbook Page 222

📖 Page 222Reflection

Reflect square ABCD along the diagonal AC and along the horizontal line. Where do the corners go?

1Diagonal AC: A and C stay; B goes to D’s place and D to B’s.

2Horizontal line: A ↔ D and B ↔ C.

✅ Answer: Diagonal: A, C fixed, B ↔ D. Horizontal: A ↔ D, B ↔ C.

📖 Page 222Ink Blot Devils

Fold a paper, drop ink on one half, press and open. Is the result symmetric? Where is the line?

1The ink is copied onto the other half like a mirror.

2The fold is the line of symmetry; usually there is no other line.

✅ Answer: Yes — the fold is the line of symmetry.

📖 Textbook Page 223

📖 Page 223Folding and cutting

A folded sheet is cut along the dotted line. Sketch the unfolded paper. Is there a line of symmetry?

1The cut is made through both layers, so the hole is copied on both sides of the fold.

2When opened, the hole is symmetric about the fold.

✅ Answer: The fold line is the line of symmetry.

📖 Textbook Page 224

📖 Page 224Punching · Q1

A hole was punched in a folded square sheet and unfolded. Identify the fold line. How was (d) folded?

Holes after unfolding.

1(a) holes left and right at the same height → vertical fold.

2(b) two holes near the top-right corner, mirror images across the diagonal from top-right to bottom-left → diagonal fold.

3(c) holes one above the other on the right → horizontal fold.

4(d) four holes at four corners from one punch → folded twice (in half vertically and then horizontally), punched near the corner where the four outer corners meet.

Fold lines in red.

✅ Answer: (a) vertical (b) diagonal (c) horizontal (d) folded into quarters.

📖 Page 224Punching · Q2

Given the line(s) of symmetry, find the other hole(s).

Given one hole and the line of symmetry.

1Each new hole is the mirror image: same distance from the line, on the other side, along a perpendicular.

2(a) near the top edge (b) top right (c) same height on the right side (d) and (e) just across the diameter at the same place.

Red holes are the mirror images across the dashed line.

✅ Answer: See the red holes.

📖 Textbook Page 225

📖 Page 225Paper cutting · Q4

Predict the shape of the hole after each cut, then check by cutting.

1(a) The zig-zag cut is at the fold, so the opened hole has the same zig-zag on both sides — a symmetric ‘bow’ shape.

2(b) The flag-shaped cut at the fold opens into a band with a V-notch at each end (like a bow tie ribbon).

3(c) Folded twice; the square cuts on folded edges open into rectangle notches as shown.

4(d) The two cuts at the edges give an I-shape (both sides cut equally).

✅ Answer: Each hole is symmetric about the fold line(s).

📖 Textbook Page 226

📖 Page 226Paper cutting · Q5

How can you get (a) a square hole in the centre, sides parallel to the edges, (b) a tilted (diamond) square hole, with folds and ONE straight cut?

1(b) Fold the square in half and in half again (the centre is now a corner). Cut straight across that corner, at 45°. Unfold: a diamond (tilted square).

2(a) Fold in half twice, then fold once more along the diagonal through the centre corner. Make one straight cut perpendicular to a folded edge near the centre corner. Unfold: a square hole with sides parallel to the edges.

3Check: all sides equal and all angles 90°.

✅ Answer: Fold to bring the centre to a corner, then one straight cut.

📖 Page 226Lines of symmetry · Q6

How many lines of symmetry: (a) the tilted square and the 8-point star (b) equilateral triangle (c) regular hexagon?

1(a) Square: 4. The 8-point star (two overlapping squares): 8.

2(b) 3 — one through each corner and the middle of the opposite side.

3(c) 6 — 3 through opposite corners and 3 through midpoints of opposite sides.

4 lines of symmetry

8 lines of symmetry

3 lines of symmetry

6 lines of symmetry

✅ Answer: (a) 4 and 8 (b) 3 (c) 6

📖 Textbook Page 227

📖 Page 227Trace · Q7

Trace each figure and draw the lines of symmetry, if any.

1Rhombus patterns: 3 rhombi (1 on top of 2) → 1 line; 3 in a row → 2; 4 in a diamond → 2; 5 in an X-pattern → 2.

2Grid figures: (a) square frame with inner square → 4 lines; (b) 8-sided figure, 4 wide and 6 tall → 2 lines; (c) pentagon → 1 line (horizontal); (d) 4-pointed star → 4 lines.

Lines of symmetry of the rhombus patterns.

(a) 4 lines

(b) 2 lines

(c) 1 line

(d) 4 lines

✅ Answer: 1, 2, 2, 2; grid figures 4, 2, 1, 4.

📖 Textbook Page 228

📖 Page 228Kolam · Q8

Find the lines of symmetry of the kolam.

1The 7 star patterns sit like the corners and centre of a regular hexagon.

2Lines through opposite outer stars: 3. Lines through the small rhombi between stars: 3.

Kolam (simplified): 6 lines of symmetry, like a regular hexagon.

✅ Answer: 6 lines of symmetry.

📖 Page 228Triangles · Q9

Draw (a) a triangle with exactly one line of symmetry (b) exactly three (c) none. Can a triangle have exactly two?

1(a) Isosceles triangle (two equal sides): 1 line.

2(b) Equilateral triangle: 3 lines.

3(c) Scalene triangle (all sides different): none.

4If a triangle had 2 lines, it would have two pairs of equal sides, so all three sides equal — then it has 3 lines.

3 lines of symmetry

✅ Answer: Isosceles, equilateral, scalene. Exactly two lines is not possible.

📖 Page 228Curved figures · Q10

Draw figures with a curved boundary having exactly (a) 1 (b) 2 (c) 4 lines of symmetry.

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

2(b) An eye/lens shape or an oval (ellipse): 2 lines.

3(c) A square with a semicircle on each side, or a 4-petal flower: 4 lines.

✅ Answer: e.g. semicircle; oval; 4-petal flower.

📖 Page 228Complete the figure · Q11

Copy and complete each figure so that the blue line is a line of symmetry.

(a)

(b)

(c)

(d)

(e)

(f)

1Take each corner of the red figure, count squares to the blue line, and mark the same distance on the other side.

2For slanting blue lines (c) and (f), count along the squares diagonally — or turn the book so the line looks vertical.

3Join the new points in the same order.

(a)

(b)

(c)

(d)

(e)

(f)

✅ Answer: See the completed figures (green dashed).

📖 Textbook Page 229

📖 Page 229Two lines · Q12

Complete each drawing so that it has both blue lines as lines of symmetry.

(a)

(b)

(c)

(d)

(e)

(f)

1Reflect the red part in the first line, then reflect both parts in the second line.

2You get 4 copies in total, one in each region between the lines.

(a)

(b)

(c)

(d)

(e)

(f)

✅ Answer: See the completed figures (green dashed).

📖 Textbook Page 230

📖 Page 230Dot grid · Q13

For each figure draw two more lines to make a shape with a line of symmetry.

(1)

(2)

(3)

(4)

(5)

(6)

1(1) Join to the dot 1 right of the top-right end: a kite with a slanting line of symmetry.

2(2), (3), (5): close the figure so it matches on both sides of a vertical line.

3(4) Close it with the two lines that make it match across the diagonal.

4(6) Close it so it matches across the horizontal middle line.

(1)

(2)

(3)

(4)

(5)

(6)

✅ Answer: See the green lines (other answers are possible).

📖 Page 2309.2 Windmill

Does the paper windmill have a line of symmetry? What happens when it is turned by 90° about the centre?

Paper windmill: looks the same after 90°, 180°, 270° and 360° turns (no line of symmetry).

1Folding never makes the halves overlap.

2A quarter turn (90°) gives exactly the same picture.

✅ Answer: No line of symmetry, but rotational symmetry (90°, 180°, 270°, 360°).

📖 Textbook Page 231

📖 Page 231Square · angles of symmetry

How many angles of symmetry does a square have? Where is the centre of rotation?

190° takes A to B’s place, B to C’s, C to D’s, D to A’s.

2180° and 270° also work, and 360° always works.

3The centre of rotation is where the diagonals cross.

Square ABCD turned 90° about its centre: A goes to B’s place, B to C’s, C to D’s, D to A’s.

✅ Answer: 4 angles: 90°, 180°, 270°, 360°; centre = where the diagonals meet.

📖 Textbook Page 232

📖 Page 232Example · Strip

Find the angles of symmetry of the strip (a trapezium-shaped strip).

1After 180° the slanting ends point the other way — it does not fit.

2Only 360° brings it back.

✅ Answer: Only 360° — no rotational symmetry.

📖 Page 232Radial arms

A figure has 4 radial arms at 90°. What are its angles of symmetry? How can it be changed to have only 2?

1Each quarter turn fits: 90°, 180°, 270°, 360°.

2Make two opposite arms different from the other two (e.g. bend two arms, or make two arms longer) — then only 180° and 360° work.

4 radial arms

Bent arms: still 4 angles (90° steps)

✅ Answer: 90°, 180°, 270°, 360°; change one opposite pair to get only 180° and 360°.

📖 Textbook Page 233

📖 Page 233Three radial arms

A figure with 3 radial arms at unequal angles: what are its angles of symmetry? How can it get 3?

1With unequal angles only 360° works.

2If ∠A = ∠B = ∠C, each = 360° ÷ 3 = 120°.

3Then 120°, 240° and 360° are angles of symmetry.

Unequal angles: only 360°

120° apart: 3 angles

✅ Answer: Unequal: only 360°. Equal (120° apart): 120°, 240°, 360°.

📖 Textbook Page 235

📖 Page 2355 and 6 arms

Draw radial-arm figures with exactly (a) 5 (b) 6 angles of symmetry and list the angles.

1(a) 5 arms, 360° ÷ 5 = 72° apart: 72°, 144°, 216°, 288°, 360°.

2(b) 6 arms, 60° apart: 60°, 120°, 180°, 240°, 300°, 360°.

5 radial arms

6 radial arms

✅ Answer: (a) multiples of 72° (b) multiples of 60°.

📖 Page 2357 arms

A radial-arm figure has exactly 7 angles of symmetry. What is the smallest angle?

1360 ÷ 7 = 51 remainder 3.

2So the smallest angle is 51 3/7 degrees — not a whole number.

✅ Answer: 51 3/7°

📖 Page 235Figure it Out · Q1

Find the angles of symmetry about the marked point: (a) plus with equal arms (b) line with a small square at each end on the same side (c) H-shape.

(a)

(b)

(c)

1(a) 90°, 180°, 270°, 360°.

2(b) After 180° the squares would be on the other side — only 360°.

3(c) 180° and 360°.

✅ Answer: (a) 90°, 180°, 270°, 360° (b) 360° only (c) 180°, 360°

📖 Page 235Figure it Out · Q2

Which figures have more than one angle of symmetry?

1Circle with 2 diameters (4), equilateral triangle (3), circle with 3 radii (3), 4-petal pinwheel (4), X of two equal lines (2 or 4), star (5): all more than one.

2Two semicircles on one side of a line: only 360°.

✅ Answer: All except the last (two semicircles).

📖 Textbook Page 236

📖 Page 236Figure it Out · Q3

Give the order of rotational symmetry of each figure.

1Line with flags pointing opposite ways: 2.

2Two lines crossing at right angles: 4 (the small arc is only an angle mark; if we count it as part of the figure, only 360° works).

36-point star: 6.

4Three bent arms: 3.

5Plus (equal arms): 4.

6Regular pentagon: 5.

✅ Answer: 2, 4, 6, 3, 4, 5

📖 Page 236Multiples of the smallest angle

Are all angles of symmetry multiples of the smallest one? True or false: every figure has 360° as an angle of symmetry.

1Turning by the smallest angle again and again gives all the others.

2A full turn (360°) always brings any figure back.

3So the smallest angle (if a whole number) divides 360.

✅ Answer: Yes; True.

📖 Textbook Page 237

📖 Page 237Circle

What are the angles of symmetry and lines of symmetry of a circle?

1Any rotation about the centre fits.

2Every diameter is a line of symmetry.

✅ Answer: Every angle; every diameter (infinitely many lines).

📖 Textbook Page 238

📖 Page 238Figure it Out · Q1

Colour sectors of a 12-sector circle to get (i) 3 angles of symmetry (ii) 4. (iii) What numbers are possible?

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

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

3(iii) The pattern must repeat evenly round 12 sectors: 1, 2, 3, 4, 6 or 12 angles (factors of 12).

3 angles

4 angles

✅ Answer: (iii) 1, 2, 3, 4, 6, 12

📖 Page 238Figure it Out · Q2

Draw two figures (not circle or square) that have both reflection and rotational symmetry.

1Equilateral triangle: 3 lines, angles 120°, 240°, 360°.

2Rectangle: 2 lines, angles 180°, 360°. Also regular hexagon, plus sign.

✅ Answer: e.g. equilateral triangle, rectangle, regular hexagon.

📖 Page 238Figure it Out · Q3

Sketch where possible: (a) a triangle with at least 2 lines and 2 angles (b) a triangle with 1 line but no rotational symmetry (c) a quadrilateral with rotational but no reflection symmetry (d) a quadrilateral with reflection but no rotational symmetry.

1(a) Equilateral triangle.

2(b) Isosceles triangle (not equilateral).

3(c) Parallelogram that is not a rectangle or rhombus (180° only).

4(d) Kite or isosceles trapezium.

✅ Answer: (a) equilateral (b) isosceles (c) parallelogram (d) kite / isosceles trapezium

📖 Page 238Figure it Out · Q4

60° is the smallest angle of symmetry. What are the others?

1Multiples of 60 up to 360.

✅ Answer: 120°, 180°, 240°, 300°, 360°

📖 Page 238Figure it Out · Q5

60° is an angle of symmetry and there are two angles of symmetry smaller than 60°. What is the smallest?

1All angles are multiples of the smallest one, s.

260 is a multiple: 60 = 3 × s gives exactly two smaller ones (s and 2s).

3s = 20°.

✅ Answer: 20°

📖 Page 238Figure it Out · Q6

Can a figure have smallest angle of symmetry (a) 45° (b) 17°?

1(a) 45 × 8 = 360 → yes (e.g. 8 radial arms).

2(b) 360 ÷ 17 is not a whole number → no.

✅ Answer: (a) Yes (b) No

📖 Textbook Page 239

📖 Page 239Figure it Out · Q7

New Parliament Building outline: (a) lines of symmetry? (b) angles of rotational symmetry?

Outline of the Parliament building.

1The outline is a triangle with its corners cut off equally (3 long and 3 short sides).

2(a) 3 lines — through the middle of each long side.

3(b) 120°, 240°, 360°.

Outline: 3 lines of symmetry; angles of symmetry 120°, 240°, 360°.

✅ Answer: (a) 3 lines (b) 120°, 240°, 360°

📖 Page 239Figure it Out · Q8–Q9

Regular polygons (triangle, square, pentagon, …): how many lines and angles of symmetry?

1A regular polygon with n sides has n lines and n angles of symmetry.

2Triangle 3, square 4, pentagon 5, hexagon 6, …

✅ Answer: 3, 4, 5, 6, 7, … — the counting numbers from 3.

📖 Page 239Figure it Out · Q10

How many lines and angles of symmetry do the Koch snowflake shapes have?

1Each stage keeps the 6-fold pattern of the star.

2Lines: 6. Angles: 60°, 120°, 180°, 240°, 300°, 360°.

✅ Answer: 6 lines and 6 angles of symmetry.

📖 Page 239Figure it Out · Q11

Lines and angles of symmetry of the Ashoka Chakra?

1It has 24 equally spaced spokes.

224 lines of symmetry; smallest angle 360° ÷ 24 = 15°.

✅ Answer: 24 lines and 24 angles (multiples of 15°).

📖 Page 239Playing with Tiles

Complete the 8 × 8 tile figure so it has exactly 2 lines of symmetry (the red lines).

Top-left quarter given; red lines must be lines of symmetry.

1Reflect the top-left quarter in the vertical red line: each tile flips left-right.

2Reflect the top half in the horizontal red line: each tile flips top-bottom.

Completed: each quarter is the mirror image across the red lines.

✅ Answer: See the completed design.

📖 Textbook Page 241

📖 Page 241Game · Winning strategy

6 × 6 grid; players cover two adjacent squares in turn; whoever cannot move loses. Is there a winning strategy?

1Let the other player start.

2Each time, draw your line at the place you get by turning their line 180° about the centre of the grid.

3Your copy never overlaps theirs (no domino is its own half-turn copy on a 6 × 6 grid), and the board stays symmetric.

4So whenever they can move, you can too — they run out of moves first.

Second player copies every move turned 180° about the centre (red dot).

✅ Answer: The second player wins by copying every move with a half turn (180°) about the centre.

🌍 Where do we see symmetry?

• Rangoli and kolam, temples and monuments, flags and logos, wheels, fans, flowers and butterflies.

🚀 Link to higher classes

• Class 7: symmetry in transformations. Class 8 and 9: reflection and rotation in coordinate geometry.

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🔑 keytoenjoylearningmaths.blogspot.com · Solutions written in our own words, based on NCERT Ganita Prakash Class 6 (Grade 6)