CLASS 6 · GANITA PRAKASH · CHAPTER 9
Explore reflection and rotational symmetry in shapes, patterns and designs.
53 worked answers · short steps first, explanations when you need them
Textbook pages 218–229
Question 1
Fold or reflect each design across candidate lines.
The flower repeats in six mirror directions, the rangoli in four, and the butterfly across its body.
Flower: 6 lines; rangoli: 4; butterfly: 1; pinwheel: 0; cloud: 0.
Question 2
Reflect each point or feature on one side across the line.
Check whether its partner lands on a matching feature at the same distance on the other side.
It is a line of symmetry only if every part matches its reflected partner; visual closeness alone is not enough.
Question square
There are two lines through the midpoints of opposite sides.
There are also two diagonal lines through opposite corners.
4 lines of symmetry: vertical, horizontal and both diagonals.
Question rectangle
Fold along a diagonal.
The adjacent side lengths are different, so the two triangular halves do not coincide.
No. A non-square rectangle has only two symmetry lines, through the midpoints of opposite sides.
Question reflection-square-vertical
The left and right vertices switch places.
A↔B and D↔C (with the labels as shown in the textbook figure).
Question reflection-square-diagonal
The diagonal endpoints lie on the mirror line, so they stay fixed.
The remaining two corners exchange places.
A and C stay fixed; B and D swap.
Question reflection-square-horizontal
Top and bottom vertices exchange with their matching partners.
A↔D and B↔C.
Question inkblot
The wet ink on one half transfers to the matching position on the other half.
The fold becomes a line of symmetry: each ink mark is reflected to an equal mark on the other side.
Question paper-cut
Keep the fold as the mirror line.
Copy each cut edge and corner at the same distance on the opposite side.
Reflect the cut outline across the fold; the unfolded hole or edge has mirror symmetry about that fold.
Question punch-1
Join matching holes with a perpendicular segment.
The fold line lies halfway between each matching pair.
The fold is the perpendicular bisector of the matching hole pairs. For a single centre hole, the paper was folded across both centre lines (vertical and horizontal).
Question punch-2
For each hole, reflect its centre across every fold line.
Keep the same perpendicular distance from the fold.
Place a mirror copy across each fold; with two folds, include all reflected positions.
Question cut-hole-square-a
Fold the sheet in half horizontally, then vertically.
At the closed corner, make a straight cut across equal distances on both folded edges.
Unfold all layers.
A central square hole is formed when the cut removes a matching quarter-square from each of the four reflected layers.
Question cut-hole-square-b
Fold horizontally and then vertically.
At the closed corner, cut diagonally so the reflected cuts form four equal sides.
Unfold.
The mirrored diagonal cuts form a centred diamond-shaped square.
Question triangle-lines
For an equilateral triangle, draw a line from each vertex to the midpoint of the opposite side.
A scalene triangle has no matching reflected halves.
Equilateral triangle: 3 lines. Scalene triangle: 0 lines.
Question hexagon-lines
Three lines join opposite vertices.
Three more pass through midpoints of opposite sides.
6 lines of symmetry.
Question kolam
Test the vertical, horizontal and diagonal candidates by reflecting each loop and endpoint.
A candidate counts only if every curve overlays its partner.
The lines that make every loop and endpoint pair coincide are the kolam’s symmetry lines; verify each by folding or tracing.
Question construct-triangles
An isosceles non-equilateral triangle has one line.
An equilateral triangle has three.
A scalene triangle has none.
Examples: isosceles (not equilateral): exactly 1; equilateral: 3; scalene: 0. Exactly 2 is impossible for a triangle.
Question curved-shapes
One: a non-circular heart shape symmetric about one vertical axis.
Two: a non-circular ellipse has vertical and horizontal axes.
Four: a rounded square with matching sides and corners.
Examples: a symmetric heart (1), an ellipse (2), and a rounded square with four matching corners (4).
Question complete-one-line
For every given point, plot its reflected partner on the other side of the blue line.
Avoid adding extra repeated structure that creates another mirror line.
Copy the given half by reflection across the blue line; choose an asymmetric half so no additional symmetry is introduced.
Question complete-two-lines
Reflect every feature across the first line.
Reflect the result across the second line as well.
Check that all four reflected copies agree where they meet.
Make the drawing invariant under reflection across both marked lines; each feature is copied into the corresponding reflected positions.
Question dot-grid
Choose the intended mirror line first.
Add the two segments as reflected copies at equal distances on opposite sides.
Many answers are possible; place each added segment as the exact mirror image of its partner across the chosen axis.
Textbook pages 230–241
Question windmill
Turn it about its centre by one quarter-turn.
The four identical blades take each other’s places.
A mirror fold reverses their handed direction instead of matching them.
It has rotational symmetry at 90°, 180°, 270° and 360°, but no reflection line.
Question square-angles
A square first matches itself after a quarter-turn: 360°÷4=90°.
Keep adding 90° until one full turn.
90°, 180°, 270° and 360°.
Question strip
Rotate it halfway and compare its ends.
The distinct ends do not exchange into a matching pattern.
No; the pictured strip matches itself only at 360°.
Question four-arms
Divide one full turn into four equal gaps: 360°÷4=90°.
List the positive multiples up to 360°.
90°, 180°, 270° and 360°.
Question three-arms
Divide 360° equally among the three arms: 360°÷3=120°.
List successive turns of 120°.
Space them by 120°; the angles are 120°, 240° and 360°.
Question five-six-arms
For five arms, the step is 360°÷5=72°.
For six arms, the step is 360°÷6=60°.
Five arms: 72°, 144°, 216°, 288°, 360°. Six arms: 60°, 120°, 180°, 240°, 300°, 360°.
Question seven-arms
Smallest turn=360°÷7.
360÷7=51 remainder 3, so 360/7=51 3/7.
51 3/7°; it is not a whole number of degrees.
Question rotations-figures
Quarter-turn: use multiples of 90°.
Half-turn: use multiples of 180°.
If it only matches at a full turn, list 360°.
Examples: quarter-turn symmetry gives 90°, 180°, 270°, 360°; half-turn gives 180°, 360°; otherwise 360° only.
Question rotation-order
Count how many times the figure matches itself during one full 360° turn.
Equivalently, divide 360° by the smallest matching angle.
Order = 360°÷smallest positive angle of symmetry. A figure with only a full-turn match has order 1.
Question multiples
Rotate by the smallest angle repeatedly.
Each repeated turn lands on another matching orientation until reaching 360°.
Yes. The angles are positive multiples of the smallest angle, up to 360°.
Question true-360
A 360° turn returns every point to its original position.
True. Every figure matches itself after a full turn.
Question true-divisor
A complete turn contains an integer number of repeats.
Thus 360°=n×(smallest angle) for a whole number n.
True. The smallest angle is a factor of 360°.
Question circle
Rotating a circle about its centre by any angle leaves it unchanged.
Every diameter divides it into two matching halves.
Every rotation angle is a symmetry angle, and every diameter is a line of reflection symmetry.
Question colour-sectors
For order 3, colour any four-sector block, then repeat it three times around the circle. The matching turn is 120°.
For order 4, colour a three-sector block and repeat it four times. The matching turn is 90°.
For 3-fold symmetry, repeat one 4-sector pattern three times. For 4-fold symmetry, repeat one 3-sector pattern four times. Possible rotational orders are 1, 2, 3, 4, 6 and 12.
Question both-symmetries
A non-square rectangle has two mirror lines and a half-turn symmetry.
An equilateral triangle has three mirror lines and 120°/240° rotations.
Examples: a non-square rectangle and an equilateral triangle.
Question triangle-symmetries
Equilateral: 3 reflection axes and 120°, 240°, 360° turns.
Non-equilateral isosceles: 1 axis and no non-full rotational symmetry.
A triangle with non-full rotational symmetry must be equilateral.
(a) Equilateral triangle; (b) non-equilateral isosceles triangle; (c) impossible for a triangle; (d) non-equilateral isosceles triangle.
Question quadrilateral-symmetries
A general parallelogram has a half-turn symmetry but no mirror axis.
A non-rhombus kite has one mirror axis but no non-full rotational symmetry.
Use a slanted non-rectangle parallelogram for rotation only, and a kite with unequal adjacent side pairs for reflection only.
Question angles-after-60
Add 60° repeatedly until reaching one full turn.
60°, 120°, 180°, 240°, 300° and 360°.
Question small-angle-from-count
The symmetry angles are equal steps.
If two positive angles are smaller than 60°, then 60° is the third step: 60°÷3=20°.
20°.
Question 45-or-17
360÷45=8, a whole number of repeats.
360÷17 is not a whole number.
45° is possible; 17° is not.
Question parliament
The three sides and corners repeat around the centre.
Three mirror axes pass through corresponding corners and opposite side midpoints.
A turn of 120° or 240° maps the boundary onto itself.
3 lines of reflection symmetry; rotational angles 120°, 240° and 360°.
Question regular-polygons
A regular n-sided polygon has one symmetry axis corresponding to each side/vertex position.
List n for n=3,4,…,10.
3, 4, 5, 6, 7, 8, 9, 10: the counting-number sequence.
Question regular-polygon-rotations
There are n matching orientations during one full turn.
The smallest angle is 360°/n.
For regular polygons from triangle to decagon, the counts are 3,4,5,6,7,8,9,10 respectively.
Question regular-polygon-sequence
A regular n-sided polygon matches itself after each turn of 360°/n.
Including the full turn, there are n matching positions.
Substitute n=3,4,…,10.
The counts are 3, 4, 5, 6, 7, 8, 9, 10, giving the counting-number sequence.
Question tile-two-lines
Choose two perpendicular lines through the centre of the design.
Place or colour each tile together with its mirror images across both lines.
Check diagonal candidates too, so the design does not gain extra symmetry lines.
Reflect the tile pattern across exactly two perpendicular centre lines, then check that no other line also maps it onto itself.
Question tile-one-line
Choose a single mirror line through the centre.
Arrange eight tile positions on one side and place their reflected partners on the other.
Choose an arrangement that is not symmetric across any second line.
Make four mirrored pairs on each side of one chosen axis (16 tiles total), using a pattern that has no second mirror axis.
Question tile-two-lines-16
Choose perpendicular horizontal and vertical centre lines.
Place four groups of four tiles that are reflections of one another across both lines.
Vary the pattern along each line to avoid diagonal symmetry.
Arrange four matching reflected groups around the centre with horizontal and vertical mirror axes, then check that diagonal reflection fails.
Question tile-creative
Choose the symmetry you want first: a mirror line, a half-turn, or several repeated rotations.
Place each tile together with the required reflected or rotated copies.
Check the whole pattern after each transformation.
Many designs are possible. State the chosen symmetry and verify that every tile maps to a tile in the same position and orientation.
Question koch
The first stage is an equilateral triangle.
Later stages keep the repeating six-part snowflake structure.
Across the pictured stages, the line counts are 3, 6, 6, 6, 6; the counts of rotational symmetry angles are also 3, 6, 6, 6, 6.
Question ashoka-chakra
The 24 identical spokes repeat every 360°÷24=15°.
Each spoke reflection axis also matches the opposite spoke arrangement.
24 reflection lines and 24 rotational symmetry angles; the smallest rotation is 15°.
Question tile-symmetry
Choose two perpendicular mirror lines through the centre.
Place each tile and colour so its reflections across both lines match.
Avoid extra matching axes if exactly two are required.
Many designs work; mirror the tile arrangement across exactly two chosen axes and check that no other line also matches.
Question domino-game
After the first player places a domino, place the domino obtained by rotating that move 180° about the board centre.
The paired placement is disjoint and legal; repeat after every move.
The second player wins by responding with the 180°-rotated copy of the first player’s move. The board is filled in move-pairs, so the first player has no move after the second player’s final response.