Symmetry — Questions and Answers
Welcome to HSLC Guru. This page gives the complete question–answer solution of Chapter 9, Symmetry, from the ASSEB (Assam State School Education Board) Class 6 New Mathematics textbook. Every “Work it Out” exercise is reproduced and answered fully, with the figures described in words.
Summary
Many objects around us — butterflies, leaves, flowers and kites — have symmetrical shapes. If a figure can be folded along a line so that one half falls exactly on the other, that line is called a line of symmetry or the axis of symmetry. A plane mirror placed on the line of symmetry makes the image overlap exactly with the part behind it, so each half is the mirror image of the other.
A figure may have one, two, three, four or infinitely many lines of symmetry. An isosceles triangle has one, a rectangle two, an equilateral triangle three, a square four and a circle infinitely many. Symmetry that occurs when an object is reflected across a line is called reflection symmetry.
When a shape is rotated about a fixed point and looks the same in certain positions, it has rotational symmetry. The fixed point is the centre of rotation. The angle at which the figure looks identical is the angle of symmetry, and the number of times the figure looks the same during one complete turn is the order of rotational symmetry.
| Figure | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Isosceles triangle | 1 | 1 |
| Equilateral triangle | 3 | 3 |
| Rectangle | 2 | 2 |
| Square | 4 | 4 |
| Regular pentagon | 5 | 5 |
| Regular hexagon | 6 | 6 |
| Circle | infinite | infinite |
Summary: This ASSEB Class 6 New Mathematics Chapter 9 (Symmetry) note explains line of symmetry and axis of symmetry, reflection symmetry, figures with one, two, three, four or infinitely many lines of symmetry, formation and use of symmetry, rotational symmetry, centre of rotation, angle of symmetry and order of rotational symmetry, with full worked answers to every “Work it Out” exercise of the chapter Symmetry.
Textbook Questions and Answers
Work it Out 9.1
1. List and draw five symmetrical objects around you.
Answer: Five symmetrical objects around us are — (a) a butterfly, (b) a tree leaf, (c) a flower, (d) a kite and (e) a star. Each of these can be folded along a line so that its two halves match exactly, so each is symmetrical. (Draw each object and mark one line of symmetry through it.)
2. Identify the line of symmetry from the lines shown in the symmetrical figure.
Answer: The figure is a house-shaped pentagon with a peak at the top. Only the vertical line passing through the peak is the line of symmetry, because folding along it makes the left and right halves match exactly. The horizontal line l₁ is not a line of symmetry.
3. Draw the lines of symmetry of the following figures.
Answer: For each figure, the line along which the two folded halves coincide is a line of symmetry. For example, an isosceles triangle has one line (from the apex to the midpoint of the base), a rectangle has two (joining midpoints of opposite sides), a square has four and a circle has a line of symmetry along every diameter. Draw the fold lines on each figure accordingly.
4. The figures are drawn on graph paper on the lines L, M and N. Complete the figures so that they are symmetrical about the given lines.
Answer: This is a drawing task. On the other side of the given line (L, M or N), draw the exact mirror image of the part already shown. Every point must be placed at the same distance on the opposite side of the line as it is on the given side; joining these mirror points completes a figure that is symmetrical about the line.
Work it Out 9.2
1. A regular pentagon (which has all sides and all angles equal) is a symmetrical figure. How many lines of symmetry does a regular pentagon have?
Answer: A regular pentagon has 5 lines of symmetry. Each line joins a vertex to the midpoint of the opposite side; since there are five vertices, there are five lines of symmetry.
Work it Out 9.3
1. Draw the lines of symmetry in the following figures. How many lines of symmetry are there in the figures given below?
Answer: Drawing the fold lines gives the following counts —
- (a) The arrow/mountain-like shape — 1 (vertical) line of symmetry.
- (b) The star-like shape pointing sideways — 1 (horizontal) line of symmetry.
- (c) The bee-like figure inside a rectangle — 1 (vertical) line of symmetry.
- (d) The plus (+) shape — 4 lines of symmetry.
- (e) The circle divided into equal sectors — 6 lines of symmetry.
- (f) The two-leaf shape — 2 lines of symmetry.
2. A few geometric figures are drawn on the graph paper. Draw the lines of symmetry in each figure.
Answer: This is a drawing task. By counting the graph squares, fold each figure mentally to find where the two halves coincide, and draw that vertical, horizontal or diagonal line as its line of symmetry.
3. Considering the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 as geometrical figures, mention which figures are symmetrical. Also write the number of lines of symmetry in each symmetric figure.
Answer: Treating the printed numerals as figures, the symmetrical ones are 0, 1, 3 and 8 —
| Numeral | Lines of symmetry |
|---|---|
| 0 | 2 (vertical and horizontal) |
| 1 | 1 (vertical) |
| 3 | 1 (horizontal) |
| 8 | 2 (vertical and horizontal) |
The numerals 2, 4, 5, 6, 7 and 9 have no line of symmetry.
4. Consider the capital letters of the English alphabet as geometric figures and prepare a list of symmetric figures. Also write the number of lines of symmetry in each symmetric figure.
| Type of symmetry | Letters | Lines of symmetry |
|---|---|---|
| Vertical line only | A, M, T, U, V, W, Y | 1 |
| Horizontal line only | B, C, D, E, K | 1 |
| Both vertical and horizontal | H, I, O, X | 2 |
The letters F, G, J, L, N, P, Q, R, S and Z have no line of symmetry. (If O is treated as a circle, it has infinitely many lines of symmetry.)
5. How many lines of symmetry are there in the given figure?
Answer: The given figure is a regular hexagon, so it has 6 lines of symmetry — three joining opposite vertices and three joining the midpoints of opposite sides.
6. Draw the following: (i) a quadrilateral with exactly one line of symmetry, (ii) a quadrilateral with exactly two lines of symmetry, (iii) a quadrilateral with exactly four lines of symmetry, (iv) a triangle with exactly three lines of symmetry.
- (i) Exactly one line of symmetry — a kite (or an isosceles trapezium).
- (ii) Exactly two lines of symmetry — a rectangle or a rhombus.
- (iii) Exactly four lines of symmetry — a square.
- (iv) Exactly three lines of symmetry — an equilateral triangle.
7. Draw a figure which has infinitely many lines of symmetry.
Answer: A circle is such a figure. Every straight line passing through its centre (every diameter) is a line of symmetry, so a circle has infinitely many lines of symmetry.
8. Some figures are drawn on graph paper. Draw and complete each of them so that the resulting figure is symmetrical about the given vertical line, horizontal line and both the lines.
Answer: This is a drawing task. For the vertical line, draw the mirror image of the left part to the right at equal distances; for the horizontal line, draw the mirror image of the top part below; and for both lines together, do both reflections so that all four parts become identical.
Work it Out 9.4
1. Find the angles of symmetry for the following figures about the point marked ‘•’.
Answer: Dividing 360° into as many equal parts as the number of times the figure looks the same gives the angles of symmetry —
- (a) The plus (+) shape — order 4; angles 90°, 180°, 270° and 360°.
- (b) The 8-spoke wheel — order 8; angles 45°, 90°, 135°, 180°, 225°, 270°, 315° and 360°.
- (c) The rectangle — order 2; angles 180° and 360°.
- (d) The equilateral triangle — order 3; angles 120°, 240° and 360°.
- (e) The four-petal flower — order 4; angles 90°, 180°, 270° and 360°.
2. Find the order of rotational symmetry for the following figures.
- (a) Four-petal flower — order of rotational symmetry 4.
- (b) Swastika — order of rotational symmetry 4.
- (c) Five-pointed star — order of rotational symmetry 5.
- (d) Letter ‘S’ — order of rotational symmetry 2.
- (e) Letter ‘Z’ — order of rotational symmetry 2.
- (f) Letter ‘H’ — order of rotational symmetry 2.
3. Draw two figures that have both reflection symmetry and rotational symmetry.
Answer: A square and an equilateral triangle are two such figures. The square has four lines of symmetry (reflection) and order of rotational symmetry 4; the equilateral triangle has three lines of symmetry and order of rotational symmetry 3.
4. Draw a quadrilateral that has both line of symmetry and rotational symmetry.
Answer: A square (or a rectangle or a rhombus) is such a quadrilateral. A square has four lines of symmetry and order of rotational symmetry 4.
5. Draw a quadrilateral with reflection symmetry but without rotational symmetry.
Answer: A kite is such a quadrilateral. It has one line of symmetry (reflection), but it looks the same only after a full 360° turn, so it has no rotational symmetry (order 1).
6. What is the order of rotational symmetry in the letter ‘S’? Does the letter ‘S’ possess lines of symmetry?
Answer: When rotated about its centre, the letter ‘S’ looks the same at 180° and 360°, so its order of rotational symmetry is 2. However, folding ‘S’ along any line does not make its halves match, so it has no line of symmetry.
7. How many lines of symmetry and angles of symmetry does the Ashoka Chakra have?
Answer: The Ashoka Chakra has 24 spokes, so it has 24 lines of symmetry and 24 angles of symmetry. Its smallest angle of symmetry is 360° ÷ 24 = 15°, and the other angles are multiples of 15° — 15°, 30°, 45°, … 360°.
8. Draw a polygon of six equal sides (regular hexagon). Find the angles of symmetry of the polygon.
Answer: A regular hexagon has order of rotational symmetry 6, so its smallest angle of symmetry is 360° ÷ 6 = 60°. Its angles of symmetry are 60°, 120°, 180°, 240°, 300° and 360°.
9. If 60° is the smallest angle of symmetry of a figure, what are the other angles of symmetry of the figure?
Answer: The other angles of symmetry are multiples of 60° — 120°, 180°, 240°, 300° and 360°. (The order of rotational symmetry of the figure is 6.)
10. Draw a figure with rotational symmetry whose smallest angle of symmetry is 45°.
Answer: If the smallest angle of symmetry is 45°, the order of rotational symmetry is 360° ÷ 45° = 8. So an eight-petal flower, a regular octagon or an 8-spoke wheel is such a figure, with angles of symmetry 45°, 90°, 135°, … 360°.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. How many lines of symmetry does a square have? (a) 2 (b) 3 (c) 4 (d) 1
Answer: (c) 4
2. What is the order of rotational symmetry of an equilateral triangle? (a) 1 (b) 2 (c) 3 (d) 4
Answer: (c) 3
3. Which figure has infinitely many lines of symmetry? (a) Square (b) Rectangle (c) Circle (d) Triangle
Answer: (c) Circle
4. How many lines of symmetry does a rectangle have? (a) 1 (b) 2 (c) 3 (d) 4
Answer: (b) 2
5. How many lines of symmetry does an isosceles triangle have? (a) 1 (b) 2 (c) 3 (d) none
Answer: (a) 1
6. One complete rotation is equal to how many degrees? (a) 90° (b) 180° (c) 270° (d) 360°
Answer: (d) 360°
7. How many lines of symmetry does the Ashoka Chakra have? (a) 12 (b) 24 (c) 36 (d) 48
Answer: (b) 24
8. What is the order of rotational symmetry of the letter ‘H’? (a) 1 (b) 2 (c) 3 (d) 4
Answer: (b) 2
9. How many lines of symmetry does a regular pentagon have? (a) 3 (b) 4 (c) 5 (d) 6
Answer: (c) 5
10. What is the smallest angle of symmetry of a regular hexagon? (a) 30° (b) 45° (c) 60° (d) 90°
Answer: (c) 60°
Fill in the Blanks
- The fixed point about which a figure rotates is called the ______. (centre of rotation)
- A square has ______ lines of symmetry. (four)
- Symmetry that occurs like a mirror image is called ______ symmetry. (reflection)
- The order of rotational symmetry of a circle is ______. (infinite)
- The angles of symmetry of an equilateral triangle are 120°, ______ and 360°. (240°)
True or False
- Every figure has at least one line of symmetry. — False (a scalene triangle has none)
- The order of rotational symmetry of a rectangle is 2. — True
- The letter ‘S’ has no line of symmetry. — True
- Every diameter of a circle is a line of symmetry. — True
- An equilateral triangle has two lines of symmetry. — False (it has three)
Short Answer Questions
1. What is a line of symmetry?
Answer: A line of symmetry (or axis of symmetry) is a line along which a figure can be folded so that its two halves match each other exactly.
2. What is meant by the order of rotational symmetry?
Answer: The order of rotational symmetry of a figure is the number of times the figure looks exactly the same during one complete rotation (360°) about its centre.
3. What is reflection symmetry?
Answer: Reflection symmetry is the type of symmetry that occurs when an object is reflected across a line, so that the image is the mirror image of the object about that line.
4. Write two uses of symmetry for beautification.
Answer: (a) Symmetry is used in the designs of floor and wall tiles; (b) it is used while drawing alpanas or rangolis and in colourful paper cut-out decorations for festive and religious occasions.
Key Terms
| Term | Meaning |
|---|---|
| Symmetry | The property by which a figure can be divided into equal matching parts |
| Line of symmetry | A line that divides a figure into two equal halves |
| Axis of symmetry | Another name for the line of symmetry |
| Reflection symmetry | Symmetry formed like a mirror image across a line |
| Rotational symmetry | Symmetry seen when a figure is rotated about a fixed point |
| Centre of rotation | The fixed point about which a figure is rotated |
| Angle of symmetry | The angle of rotation at which the figure looks the same |
| Order of rotational symmetry | Number of times a figure looks the same in one full rotation |
| Isosceles triangle | A triangle with two equal sides (one line of symmetry) |
| Equilateral triangle | A triangle with all three sides equal (three lines of symmetry) |
| Regular polygon | A polygon with all sides and all angles equal |
| Radial arm | An arm extending outward from the centre of a figure |