CBSE Class 6 Maths Symmetry Worksheet Set 02

Short Answer Questions

Question 1. Place the capital letters A, B, H and N under the correct heading, and give a reason for the letter N: letters with only a vertical line of symmetry, letters with only a horizontal line of symmetry, letters with both, letters with none.
Answer:
Only a vertical line of symmetry: A
Only a horizontal line of symmetry: B
Both a vertical and a horizontal line: H
No line of symmetry: N
Reason for N: its middle stroke slants from the bottom left to the top right. Folding N from side to side or from top to bottom never makes the two parts cover each other.
In simple words: A folds left-right, B folds top-bottom, H folds both ways and N folds neither way.

Teacher's Note:
1. Test each letter with a vertical fold and a horizontal fold.
2. N has rotational symmetry but no line of symmetry.

Question 2. Give one example of each of the following, chosen from shapes or things you know:
(i) a figure with exactly one line of symmetry,
(ii) a figure with exactly four lines of symmetry,
(iii) a figure with no line of symmetry. Give a reason for your answer to (iii).
Answer:
(i) A triangle with only two equal sides. The fold through the top corner and the middle of the base makes its halves overlap.
(ii) A square: two folds through the middles of opposite sides and two folds along the diagonals.
(iii) A triangle with all three sides of different lengths. No two sides are alike, so no fold makes the two parts cover each other.
In simple words: An isosceles triangle has 1 line, a square has 4, and a scalene triangle has none.

Teacher's Note:
1. Other answers are fine if they are correct, like the letter A for (i).
2. Always give the reason asked for in (iii).

Question 3. Nina folds a figure along a straight line and finds that the part on the left does not fit exactly over the part on the right, although both parts are of the same size. She says the fold line is still a line of symmetry because the two parts are equal. Is she correct? Explain what a line of symmetry really needs.
Answer:
No, Nina is not correct.
A line of symmetry is not just a line that makes two parts of the same size.
The two parts must fall exactly on each other when the figure is folded along the line, with every corner landing on a matching corner.
Here the parts do not overlap, so the fold line is not a line of symmetry. The diagonal of a rectangle that is not a square is a well-known example of such a line.
In simple words: Equal size is not enough; the folded parts must cover each other exactly.

Teacher's Note:
1. This is the most common mistake in this chapter.
2. The rectangle's diagonal is the best example to quote.

Question 4. A four-sided figure has all its sides equal but it is not a square.
(i) How many lines of symmetry does it have, and which lines are they?
(ii) What is its order of rotational symmetry?
Answer:
The figure is a rhombus.
(i) It has 2 lines of symmetry: its two diagonals. Folding along a diagonal makes the two triangles cover each other, but a fold through the middles of opposite sides does not.
(ii) A half turn about the point where the diagonals cross brings it back onto itself. Its angles of symmetry are 180° and 360°, so its order of rotational symmetry is 2.
In simple words: A rhombus folds along its two diagonals and looks the same after a half turn.

Teacher's Note:
1. A rhombus is the opposite of a rectangle: diagonals work, middle lines do not.
2. Order 2 means angles 180° and 360°.

Question 5. Describe how you can use a small mirror to check whether a given dotted line is a line of symmetry of a figure. What will you see if the line is not a line of symmetry?
Answer:
Stand the mirror upright on the figure, with its edge exactly along the dotted line.
Look at the half of the figure on your side together with its image in the mirror.
If what you see is exactly the whole figure, the dotted line is a line of symmetry.
If it is not a line of symmetry, the mirror picture does not match the hidden half: part of the picture looks wrong or a piece sticks out.
In simple words: Put a mirror on the line; if half plus its image gives the whole figure, the line is a line of symmetry.

Teacher's Note:
1. The mirror shows the reflection of one half.
2. Keep the mirror exactly upright on the line.

Question 6. A sheet of paper is folded once and a shape is cut out through both the layers together. Explain why the opened sheet always has a line of symmetry, and say where that line lies.
Answer:
The cut passes through both layers at once, so exactly the same piece is removed from each layer.
The two layers are mirror halves of each other about the fold, so on opening, the cut on one side is the mirror image of the cut on the other side.
So the opened sheet always has a line of symmetry, and that line is the fold line itself.
In simple words: Cutting both layers together makes the two sides mirror images about the fold.

Teacher's Note:
1. This is how paper snowflakes and cut-outs are made.
2. The fold always becomes the mirror line.

Question 7. The circle is often called the most symmetric of all figures. Give one reason about folding and one reason about turning that support this.
Answer:
Folding: fold the circle along any diameter and the two halves cover each other. A circle has countless diameters, so it has countless lines of symmetry.
Turning: whatever angle the circle is turned through about its centre, it looks exactly the same. So every angle is an angle of symmetry.
In simple words: Every diameter is a fold line and every turn leaves a circle unchanged.

Teacher's Note:
1. No other figure has endless lines and endless angles.
2. Mention 'diameter' and 'centre' in the answer.

Question 8. Describe two ways in which you can test whether a given figure has rotational symmetry, without simply guessing.
Answer:
First way: trace the figure on thin paper and cut the tracing out. Place it exactly over the figure, hold the centre with a pin and turn it slowly. If it fits on the figure before a full turn is made, the figure has rotational symmetry, and that turn is an angle of symmetry.
Second way: look at the figure, then turn the book or photograph itself about the centre and see whether the picture looks exactly as it did before a full turn is completed.
In simple words: Turn a traced copy on a pin, or turn the page itself, and look for a match before a full turn.

Teacher's Note:
1. The tracing method gives exact angles.
2. Only turns smaller than 360° count.

Question 9. A paper cut-out has order of rotational symmetry 15.
(i) Find its smallest angle of symmetry.
(ii) Write all its angles of symmetry that are less than 100°.
Answer:
(i) Order 15 means the cut-out matches itself 15 times in one full turn, so the smallest angle = 360° ÷ 15 = 24°.
(ii) Angles of symmetry are multiples of 24°: 24°, 48°, 72°, 96°, 120°, … Those less than 100° are 24°, 48°, 72° and 96°.
In simple words: 360° ÷ 15 = 24°, and the multiples of 24° below 100° are 24°, 48°, 72° and 96°.

Teacher's Note:
1. Smallest angle = 360° ÷ order.
2. Stop listing once you pass 100°.

Question 10. Meena folds a rectangular sheet that is not a square in four different ways: along the line joining the middles of the left and right sides, along the line joining the middles of the top and bottom sides, and along each of the two diagonals. For each fold, say whether the two parts overlap exactly, and hence write how many lines of symmetry the sheet has.
Answer:
Fold along the line joining the middles of the left and right sides: the parts overlap exactly.
Fold along the line joining the middles of the top and bottom sides: the parts overlap exactly.
Fold along either diagonal: the parts do not overlap; the long side of one triangle falls on the short side of the other and pieces stick out.
So the sheet has only 2 lines of symmetry, and neither diagonal is one of them.
In simple words: Only the two middle folds work, so the rectangle has 2 lines of symmetry.

Teacher's Note:
1. Actually folding the sheet makes this clear.
2. Diagonals fail because the sides are of different lengths.

Question 11. A figure is reflected in a vertical line of symmetry drawn through it. Explain what happens to
(i) a point that lies on the line itself,
(ii) a point that lies 3 units to the left of the line.
Answer:
(i) A point on the line does not move. Its distance from the line is zero, so its mirror image is the point itself.
(ii) It goes to the point 3 units to the right of the line, at the same level. The line of symmetry is always exactly midway between a point and its image.
In simple words: Points on the mirror line stay put; other points jump to the same distance on the other side.

Teacher's Note:
1. Same distance, opposite side, same level.
2. This is how we complete half-drawn figures.

Question 12. A design is made of identical arms coming out from a centre point, all placed at equal angles. Its smallest angle of symmetry is 36°.
(i) How many arms does the design have?
(ii) Write its order of rotational symmetry and its third smallest angle of symmetry.
Answer:
(i) The arms share a full turn equally, so number of arms = 360° ÷ 36° = 10.
(ii) The design matches itself after every 36°, so its order of rotational symmetry is 10. The angles of symmetry are 36°, 72°, 108°, …, so the third smallest is 3 × 36° = 108°.
In simple words: 10 arms, order 10, and the third angle is 108°.

Teacher's Note:
1. Number of equal arms = order of rotational symmetry.
2. The k-th angle is k × smallest angle.

Question 13. Differentiate between a line of symmetry and an angle of symmetry of a figure. Write two points of difference and give one example for each.
Answer:
1. A line of symmetry is a line: when the figure is folded or reflected about it, the two parts overlap exactly. An angle of symmetry is an amount of turn about a fixed centre after which the figure looks the same.
2. A line of symmetry is tested by folding or with a mirror; an angle of symmetry is tested by turning a traced cut-out about the centre.
Examples: the letter A has one line of symmetry (vertical) but no angle of symmetry other than 360°. A paper pinwheel has no line of symmetry, but 90° is an angle of symmetry for it.
In simple words: A line of symmetry is about folding; an angle of symmetry is about turning.

Teacher's Note:
1. Use 'fold' for lines and 'turn' for angles.
2. Pick examples that have one kind but not the other.

Question 14. Explain why a parallelogram that is not a rectangle and not a rhombus has no line of symmetry at all, and write all its angles of symmetry.
Answer:
No fold makes its two parts cover each other:
Folding along a line joining the middles of opposite sides leaves slanting pieces sticking out on both sides.
Folding along a diagonal also fails, because the two triangles are turned the wrong way round with respect to each other.
So it has no line of symmetry.
But a half turn about the point where the diagonals cross brings it back onto itself, so its angles of symmetry are 180° and 360°.
In simple words: No fold works for this parallelogram, but a half turn does.

Teacher's Note:
1. Check middle-line folds and diagonal folds.
2. Its order of rotational symmetry is 2.

Question 15. Describe one figure that has 5 lines of symmetry and one figure that has 5 angles of symmetry but not a single line of symmetry. How does the second figure differ from the first?
Answer:
5 lines of symmetry: a regular pentagon (five equal sides and equal angles). Each line passes through a corner and the middle of the opposite side. It also has order of rotational symmetry 5.
5 angles of symmetry but no line: a paper pinwheel with 5 identical blades, all bent the same way.
Difference: the pinwheel's blades all curve in one direction, so in a mirror they would curve the other way and no fold works. The pentagon's two halves match in a mirror.
In simple words: A regular pentagon has 5 lines; a 5-blade pinwheel has 5 turns but no line because its blades all curve one way.

Teacher's Note:
1. Curved or bent parts all in one direction remove line symmetry.
2. Both figures have order 5.

Question 16. A window grill is made of a square frame with one straight bar fixed inside it, joining the middle of the left side to the middle of the right side.
(i) How many lines of symmetry does the grill have?
(ii) What is its order of rotational symmetry?
Answer:
(i) 2 lines: the horizontal line through the middle (along which the bar lies) and the vertical line through the middle. The diagonals no longer work, because the bar does not fall on itself when folded along a diagonal.
(ii) A half turn about the centre takes the bar onto itself and the frame onto itself, but a quarter turn makes the bar vertical. So the angles of symmetry are 180° and 360°, and the order is 2.
In simple words: The bar leaves only 2 lines and a half-turn symmetry of order 2.

Teacher's Note:
1. Adding a part can remove some symmetries.
2. Check each of the square's 4 lines with the bar in place.

Question 17. A craft design has 4 identical arms coming out from a centre point at right angles, so it has 4 angles of symmetry. Describe one change you can make to this design so that it has exactly 2 angles of symmetry, and write those two angles.
Answer:
Keep the four arms at right angles, but make one pair of opposite arms long and the other pair short. (Colouring one pair of opposite arms differently also works.)
Now a quarter turn of 90° takes a long arm to the place of a short arm, so the design does not look the same.
A half turn of 180° takes each long arm to the opposite long arm and each short arm to the opposite short arm, so the design looks the same.
The two angles of symmetry are 180° and 360°.
In simple words: Make opposite arms match in pairs but differ from the other pair; then only 180° and 360° work.

Teacher's Note:
1. Opposite arms must still match each other.
2. Other correct changes are also acceptable.

Question 18. For a certain figure, both 180° and 120° are angles of symmetry.
(i) Show that 60° must also be an angle of symmetry. (Hint: turn the figure through 180° and then turn it back through 120°.)
(ii) What is the largest value that its smallest angle of symmetry can have, and what is the order of rotational symmetry then?
Answer:
(i) Turn the figure through 180°; it looks the same. Now turn it back through 120°; it still looks the same. The two turns together are one turn of 180° − 120° = 60°, after which the figure looks the same. So 60° is an angle of symmetry.
(ii) Every angle of symmetry is a multiple of the smallest one, so the smallest angle must divide 60°. The largest it can be is 60°, and then the order is 360° ÷ 60° = 6.
In simple words: 180° forward and 120° back make 60°, so the smallest angle is at most 60° with order 6.

Teacher's Note:
1. Combining two angles of symmetry gives another one.
2. The smallest angle divides every angle of symmetry.

Question 19. A regular polygon with n sides has n lines of symmetry and order of rotational symmetry n.
(i) Write the number of lines of symmetry and the smallest angle of symmetry of a regular polygon with 12 sides.
(ii) A regular polygon has 24° as its smallest angle of symmetry. How many sides does it have?
Answer:
(i) It has 12 lines of symmetry and order 12, so its smallest angle of symmetry is 360° ÷ 12 = 30°.
(ii) Order = 360° ÷ 24° = 15. For a regular polygon the order equals the number of sides, so it has 15 sides.
In simple words: 12 sides give 12 lines and 30°; 24° gives 360 ÷ 24 = 15 sides.

Teacher's Note:
1. For regular polygons: sides = lines = order.
2. Smallest angle × sides = 360°.

Question 20. Two identical square tiles are joined side by side to make a longer rectangle.
(i) How many lines of symmetry and how many angles of symmetry does this rectangle have?
(ii) One of the two square tiles is now painted black and the other is left white. How do your answers to part (i) change? Give a reason.
Answer:
(i) The rectangle is twice as long as it is high. It has 2 lines of symmetry (the long middle line and the short middle line) and 2 angles of symmetry, 180° and 360°.
(ii) The short middle line (the join of the tiles) now swaps the black tile with the white tile, so it is no longer a line of symmetry. The long middle line still takes black onto black and white onto white, so it stays. Now there is only 1 line of symmetry.
A half turn also swaps the two tiles, so 180° is lost. Only 360° is left, so the painted rectangle has no rotational symmetry.
In simple words: Painting one tile black leaves just one line of symmetry and removes the half-turn symmetry.

Teacher's Note:
1. Colour matters: black must land on black.
2. Check each old symmetry again after the change.