This MCQ module is based on: Symmetry Exercises and Summary
TOPIC 36 OF 41
Symmetry Exercises and Summary
🎓 Class 6
Mathematics
CBSE
Theory
Ch 9 — Symmetry
⏱ ~30 min
🌐 Language: [gtranslate]
🧠 AI-Powered MCQ Assessment
▲
📐 Maths Assessment
▲
This mathematics assessment will be based on: Symmetry Exercises and Summary
Targeting Class 6 level in Geometry, with Basic difficulty.
Upload images, PDFs, or Word documents to include their content in assessment generation.
9.6 Exercises — End of Chapter
Q1. In each of the following, identify the line(s) of symmetry, if any:
(a) An equilateral triangle (b) Letter E (c) Parallelogram (non-rhombus) (d) Isosceles trapezium (e) Regular hexagon.
(a) An equilateral triangle (b) Letter E (c) Parallelogram (non-rhombus) (d) Isosceles trapezium (e) Regular hexagon.
(a) 3 lines — each median/altitude. (b) 1 horizontal line through the middle. (c) 0 lines — parallelograms other than rectangles/rhombi have no line symmetry. (d) 1 line — perpendicular bisector of the two parallel sides. (e) 6 lines — three through opposite vertices and three through midpoints of opposite sides.
Q2. State the order of rotational symmetry for each: (i) parallelogram (non-rectangular) (ii) rhombus (iii) letter I (iv) letter N (v) letter T (vi) 6-petal flower with identical petals.
(i) 2 (ii) 2 (iii) 2 (iv) 2 (v) 1 — only 360° matches (vi) 6.
Q3. Determine both the number of lines of symmetry and the order of rotational symmetry for the shapes: (a) semi-circle (b) regular octagon (c) 3-blade pinwheel (identical blades) (d) letter Z.
(a) 1 line, order 1 (semi-circle has a vertical axis but no non-trivial rotation). (b) 8 lines, order 8. (c) 0 lines, order 3 (blades curve in one direction). (d) 0 lines, order 2.
Q4. Find lines of symmetry of the digits 0–9 (standard printed form).
0 → 2 lines; 1 → 1 (vertical); 2 → 0; 3 → 1 (horizontal); 4 → 0; 5 → 0; 6 → 0; 7 → 0; 8 → 2 lines; 9 → 0.
Q5. Draw the following:
(a) a triangle with exactly one line of symmetry,
(b) a triangle with exactly three lines of symmetry,
(c) a triangle with no line of symmetry.
Is it possible to draw a triangle with exactly two lines of symmetry? Justify.
(a) a triangle with exactly one line of symmetry,
(b) a triangle with exactly three lines of symmetry,
(c) a triangle with no line of symmetry.
Is it possible to draw a triangle with exactly two lines of symmetry? Justify.
(a) Isosceles triangle (two equal sides). (b) Equilateral triangle. (c) Scalene triangle.
Two lines is impossible. If a triangle has 2 lines of symmetry, each axis forces a pair of sides to be equal, meaning all three sides must be equal — which gives 3 axes, not 2. So a triangle has 0, 1, or 3 lines of symmetry — never exactly 2.
Two lines is impossible. If a triangle has 2 lines of symmetry, each axis forces a pair of sides to be equal, meaning all three sides must be equal — which gives 3 axes, not 2. So a triangle has 0, 1, or 3 lines of symmetry — never exactly 2.
Q6. An angle of rotational symmetry is 36°. What is the order? Give one figure that shows this.
Order = 360°/36° = 10. A regular 10-sided polygon (decagon), or a rangoli with 10 identical petals.
Q7. Copy the half-figure given and complete it so that the dashed vertical line is a line of symmetry. (Typical NCERT half-figures: half-star, half-tree, half-arrow.)
For every corner point P on the given half, mark its reflection P' at the same perpendicular distance on the other side of the dashed line. Connect the reflections in the same order to complete the figure. The finished figure will be identical on both sides of the dashed axis.
Q8. Identify all lines of symmetry in a traditional kolam that has 4-fold rotational symmetry. How many lines must it have?
If the kolam has 4 lines of symmetry, they are two perpendicular midlines + two diagonals (like a square). A kolam with 4-fold rotational symmetry might have 4 lines of symmetry — but some designs twist (like a swirl), giving only order 4 rotation with 0 lines.
Q9. A regular polygon has 10 lines of symmetry. What is its order of rotational symmetry and angle of symmetry? How many sides does it have?
A regular polygon with \(n\) sides has \(n\) lines of symmetry. So \(n = 10\). Order = 10, angle = 360°/10 = 36°. Shape: regular decagon.
Q10. Which letters of "MATHS" have (i) line symmetry, (ii) rotational symmetry?
(i) M (vertical axis), A (vertical), T (vertical), H (two lines). S has no line symmetry.
(ii) Only H (order 2) and S (order 2) have rotational symmetry. M, A, T have order 1.
(ii) Only H (order 2) and S (order 2) have rotational symmetry. M, A, T have order 1.
Chapter Summary
Symmetry
A figure has symmetry if a part of it repeats in a definite pattern — via folding (line) or turning (rotation).
Line Symmetry
A line along which the figure can be folded so the two halves coincide. The line is the axis of symmetry.
Reflection Rule
For any point P, the axis is the perpendicular bisector of PP' where P' is its mirror image.
Rotational Symmetry
A figure has rotational symmetry if it matches itself after a turn by some angle less than 360° about a fixed centre.
Order & Angle
Order = number of matches in a full turn. Angle of symmetry = 360° ÷ order.
Regular Polygons
A regular n-gon has n lines of symmetry and rotational order n.
Key Terms
Symmetry • Line of symmetry • Axis • Reflection • Mirror image • Centre of rotation • Order of rotational symmetry • Angle of symmetry • Regular polygon
Chapter-End Project: Design Your Own Rangoli
L6 CreateMaterials: A3 paper or cloth, coloured chalk / rangoli powder / markers, ruler, protractor
Goal: Create a symmetrical rangoli with chosen symmetry properties.
- Choose a target: (a) 4 lines of symmetry + order 4, or (b) only rotational symmetry of order 6 without any line of symmetry.
- Mark a centre on the paper.
- For target (a): draw two perpendicular lines + two diagonals dividing the space into 8 equal wedges. Design just one wedge, then reflect and rotate it into the others.
- For target (b): divide the area into 6 equal wedges (60° each). Inside each wedge, draw the same curved motif pointing clockwise. Do not reflect.
- Colour the rangoli. Show it to a classmate — can they identify its symmetry properties?
Rubric: Centre clearly marked (2), correct number of axes/rotational order shown (4), design elements match on all axes/rotations (3), creativity of motif (1). Total 10.
Competency-Based Questions
Scenario: A tile-layout designer is tiling a square-shaped hall with 400 identical triangular tiles. Each tile is a right-angled isosceles triangle. She wants the final pattern to have 4-fold rotational symmetry and 4 lines of symmetry.
Q1. A right-angled isosceles triangle has how many lines of symmetry and what rotational order?
L3 Apply1 line of symmetry (the altitude from the right angle bisecting the hypotenuse) and order 1 (no non-trivial rotation).
Q2. Analyse how the designer can combine 4 such triangles into a single square unit that has full 4-fold symmetry.
L4 AnalysePlace 4 triangles with their right-angled vertices at a common centre and hypotenuses along the outer edges. The result is a square with 4 lines of symmetry (2 diagonals + 2 midlines) and rotational order 4.
Q3. Evaluate whether 400 triangular tiles are sufficient. Can the pattern be exactly 10 × 10 repeats of the 4-triangle square unit?
L5 EvaluateYes. 4 triangles per square unit × 100 units = 400 tiles, which fits a 10 × 10 grid. The overall layout has 4 lines of symmetry (through the centre) and order 4 rotational symmetry. The count works exactly.
Q4. Create an alternative tile arrangement that uses the same 400 triangular tiles but produces only rotational order 2 (no 4-fold rotation). Describe one.
L6 CreateGroup tiles into rectangular units of 2 triangles (forming a long rectangle of 1:2 ratio). Stack 200 such rectangles in a 10 × 20 grid all aligned the same way. The whole pattern now has rotational symmetry order 2 (matches at 180°), 2 lines of symmetry (horizontal + vertical midlines), but not 4-fold.
Assertion–Reason Questions
Assertion (A): A triangle can never have exactly two lines of symmetry.
Reason (R): A triangle has at most 3 sides and each axis pairs sides.
Reason (R): A triangle has at most 3 sides and each axis pairs sides.
(a) Two axes force all 3 sides equal, giving 3 axes. So a triangle has 0, 1, or 3 lines — never exactly 2. R correctly explains A.
Assertion (A): A regular polygon with 12 sides has rotational symmetry at angles 30°, 60°, 90°, …, 330°.
Reason (R): Each multiple of 360°/12 = 30° leaves the figure unchanged.
Reason (R): Each multiple of 360°/12 = 30° leaves the figure unchanged.
(a) A regular 12-gon matches itself at every 30° rotation — 11 proper rotations + trivial 360° = order 12. R is the correct reason.
Assertion (A): A circle has the highest possible order of rotational symmetry among plane figures.
Reason (R): Any rotation of a circle about its centre leaves it unchanged.
Reason (R): Any rotation of a circle about its centre leaves it unchanged.
(a) The circle has infinite order (every angle works), and the reason correctly explains this — because every point of the circle lies at the same distance from the centre.
Frequently Asked Questions
What is the summary of Chapter 9 Symmetry?
Chapter 9 introduces line symmetry (fold test), rotational symmetry (order and angle of rotation) and how the two combine in regular polygons and everyday patterns.
How do you find the order of rotation in exercises?
Rotate the figure step by step through 360 degrees; count how many times it looks exactly like the original. That count is the order of rotation.
What are the common mistakes in symmetry problems?
Drawing diagonals of a rectangle as lines of symmetry (they aren't), confusing order of rotation with angle, and missing that some letters have both horizontal and vertical symmetry.
How many lines of symmetry does a regular polygon with n sides have?
A regular polygon with n sides has exactly n lines of symmetry and rotational symmetry of order n, with angle of rotation 360/n degrees.
Is symmetry important for higher classes?
Yes. Symmetry ideas extend to Class 7 transformations, Class 9 congruence, Class 10 coordinate geometry reflections and are used in art, design and physics.
AI Tutor
Mathematics Class 6 — Ganita Prakash
Ready