✅ SOLUTIONS
Class 5
Chapter 10: Symmetrical Designs
Math Mela · NCERT Solutions
Class 5Math Mela (NCERT)Chapter 10
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✂️ Alphabet Cutout
Which letters (E N X T K V O) can be made with a half or quarter fold? Which have horizontal/vertical/both lines of symmetry?
Solution
| Letter | Symmetry | Fold needed |
|---|---|---|
| E | Horizontal line only | Half (1/2) |
| N | Neither (only rotational!) | Cannot be made this way |
| X | Both horizontal & vertical | Quarter (1/4) |
| T | Vertical line only | Half (1/2) |
| K | Horizontal line only | Half (1/2) |
| V | Vertical line only | Half (1/2) |
| O | Both horizontal & vertical | Quarter (1/4) |
Horizontal line of symmetry: E, K, X, O
Vertical line of symmetry: T, V, X, O
Both horizontal AND vertical: X, O
Vertical line of symmetry: T, V, X, O
Both horizontal AND vertical: X, O
🎡 Windmill Firki & Letters
Does the firki look the same after 1/4, 1/2, 3/4, and a full turn?
Solution
Yes! Since a firki has 4 identical arms evenly spaced, it looks exactly the same after EVERY quarter turn (1/4, 1/2, 3/4, and full turn) — this means it has rotational symmetry of order 4.
Do letters look the same when turned? (H example given)
Solution
The letter H has rotational symmetry — it looks the same after a 1/2 turn (rotating it 180° gives back the same H shape). Other letters with this property include S, Z, N, and I.
🔢 Symmetry in Digits
Which digits have reflection/rotational/both symmetries? What about || and |00|?
Solution
Reflection symmetry: 0, 3, 8 (and simple forms of 1)
Rotational symmetry: 0, 8 (and simple forms of 1)
Both: 0, 8
"||" (like 11) and "|00|" (like 1001): Both have reflection symmetry (they read the same left-right, since each digit is itself symmetric and the pattern is a palindrome) AND rotational symmetry (turning 180° gives back the same number) — so BOTH symmetries!
Examples using symmetric digits (0, 1, 8):
| Digits | Example | Symmetry type |
|---|---|---|
| 2-digit | 11, 88 | Both |
| 3-digit | 808, 101 | Both |
| 4-digit | 8008, 1001 | Both |
| Rotational-only example | 69 (rotates to look like 69 again, since 6↔9) | Rotational only |
🎨 Making Designs
Pinwheel design (3 triangles + 1 circle): rotational symmetry? How to fix for 1/2 and 1/4 turn symmetry?
Solution
(a) No — the design does NOT have rotational symmetry. Three arms are triangles (top, left, bottom) but the fourth (right) is a circle — this breaks the pattern.
(b) For 1/2 turn symmetry: the opposite arms (top↔bottom, left↔right) must match. Top/bottom already match (both triangles). ADD a circle on the LEFT side too (matching the existing right circle) — now top=bottom(triangles) and left=right(circles) ✓
(c) For 1/4 turn symmetry: ALL FOUR arms must be identical. ADD a triangle on the right side (alongside/replacing the circle idea) so all four arms — top, right, bottom, left — are the same triangle shape, like a proper 4-armed pinwheel.
(d) Yes, both new designs would also have reflection symmetry: the 1/2-turn version has 2 lines of symmetry (horizontal and vertical, since opposite arms match); the 1/4-turn version (all 4 triangles) has 4 lines of symmetry (horizontal, vertical, and both diagonals)!
🤔 Let Us Think — Square Design
2×2 coloured square (green/pink checkerboard) — same after 1/2 or 1/4 turn?
Solution
After 1/2 turn (180°): YES, looks the same! (Top-left green ↔ Bottom-right green swap places but match; Top-right pink ↔ Bottom-left pink swap but match too)
After 1/4 turn (90°): NO, does NOT look the same (a green square would rotate into a position where the original had pink)
Colour the 8-triangle square so it looks the same after every 1/4 turn. How many times does it look the same in a full turn?
Solution — Method
The square is divided into 8 triangular sections (by the horizontal, vertical, and both diagonal lines meeting at the centre). To get 1/4-turn (90°) rotational symmetry using 2 colours: colour alternating triangles — every other triangle going around — using Colour 1, and the remaining ones Colour 2. This creates a pinwheel-like alternating pattern that repeats every 90°.
This shape looks the same 4 times during a full 360° turn (at 90°, 180°, 270°, and 360°) — rotational symmetry of order 4.
Reflection symmetry: Yes — this alternating 8-triangle pattern also has 4 lines of reflection symmetry (along the horizontal, vertical, and both diagonal fold lines)!
🔷 Squares & Triangles Shape
2×2 blue squares with tan triangles left & right — reflection symmetry? Rotational symmetry?
Solution
Reflection symmetry:YES — it has 2 lines of symmetry: one vertical line (down the middle, between the two blue squares) and one horizontal line (between the top and bottom rows).
Rotational symmetry:YES, at a 1/2 turn (180°) — rotating the whole shape 180° swaps top-left↔bottom-right and top-right↔bottom-left, and since the pattern is symmetric, it looks the same. (It does NOT have 1/4-turn symmetry, since the shape is wider than it is tall.)
Both symmetries?YES — this shape has both reflection AND rotational symmetry!
🧵 Block Printing Match
Match each wooden block (i-v) to its print (a-e).
Solution
📎 Wooden blocks (i-v) and their prints (a-e)
| Block | Matches Print | Why |
|---|---|---|
| (i) Paisley teardrop | (c) | Matching elongated paisley outline |
| (ii) Lotus flower | (a) | Matching pointed-petal bloom shape |
| (iii) Round mandala | (d) | Matching radial floral pattern |
| (iv) Fan/feather | (e) (given) | Matching fan shape |
| (v) Flowering vine | (b) | Matching elongated stem-with-leaves shape |
🌸 Design A & B — Border Shapes
📎 Design A (fern cross) and Design B (4-petal flower)
Design B looks the same after every ___ turn? What symmetry?
Solution
Design B (the 4-petal flower/clover) looks the same after every 1/4 turn — just like Design A! This design has rotational symmetry (order 4), since it has 4 identical petals arranged evenly around the centre.
Which border shapes have reflection symmetry (✓) or rotational symmetry (*)?
Solution — Method
Scan each border shape and ask two questions:
- Reflection (✓): Can I fold it in half along SOME line and both sides match? (e.g., a 6-pointed star, a regular hexagon, a heart shape — YES)
- Rotational (*): Can I turn it less than 360° and have it look the same? (e.g., a pinwheel, a 3-petal cluster, a regular hexagon or star — YES)
Shapes like regular stars, hexagons, and pentagons usually have BOTH symmetries (✓ and *). A pinwheel shape typically has ONLY rotational symmetry (*) but not reflection. An irregular or single-direction shape (like a plain parallelogram or a single arrow) often has NEITHER.