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Symmetry: questions with answers

Class 6 Maths, Chapter 9 (NCERT). 15 questions by concept, 14 with pictures. Open “Show answer” for the answer and why.

How to teach this chapter · Common misconceptions & worked visual for this chapter

Lines of Symmetry and Their Counts

Question 1

The figure shows a circle with centre O and four straight lines marked A, B, C and D. Which of these lines are lines of symmetry of the circle?

ABCDO?
  1. (a)The line D only
  2. (b)The line A only
  3. (c)The lines A, B and C only
  4. (d)The lines A and C only
Show answer

(d) The lines A and C only

A line of symmetry of a circle must pass through its centre. Lines A and C pass through the centre O, so folding along either one makes the two halves overlap exactly. Line B stays above the centre and misses O. Line D meets the circle at only one point. So the lines of symmetry shown are A and C.

Question 2

The figure shows the digits 0 through 9 in block style. Look only at the digits from 2 to 9. Which list gives the digits with no line of symmetry?

0123456789?
  1. (a)3
  2. (b)None of these digits
  3. (c)2, 4, 5, 6, 7, 9
  4. (d)8
Show answer

(c) 2, 4, 5, 6, 7, 9

Test each digit from 2 to 9 in the block style shown. Fold it and check for exact overlap. The digit 3 folds along a horizontal line. The digit 8 folds along a horizontal line and a vertical line. The digits 2, 4, 5, 6, 7 and 9 have no line of symmetry. So the digits with no line of symmetry make the list 2, 4, 5, 6, 7, 9.

Question 3

The stepped shape has a dotted line through its middle. How can you tell whether this line is a line of symmetry?

?stepped shape
  1. (a)Check that the two parts have the same area
  2. (b)Check that the line passes through the middle of the shape
  3. (c)Fold along the line and check that the two parts overlap exactly
  4. (d)Check that the line passes through two corners
Show answer

(c) Fold along the line and check that the two parts overlap exactly

Folding must make both halves overlap exactly. A middle line alone does not guarantee this. For this figure, the upper step and the lower step do not match after folding.

Symmetry as Repeated Pattern

Question 4

The figure shows a cloud-like outline. Its lobes are of different sizes and spacing. Give the correct reason that tells whether this figure is symmetrical.

?
  1. (a)It is symmetrical because it looks attractive.
  2. (b)It is not symmetrical because its parts do not repeat in a definite pattern.
  3. (c)It is not symmetrical because its outline is curved.
  4. (d)It is symmetrical because a few small bumps look similar.
Show answer

(b) It is not symmetrical because its parts do not repeat in a definite pattern.

A figure is symmetrical when its parts repeat in a definite pattern. The cloud-like outline has lobes of different sizes and spacing, so no part repeats in a definite pattern. A curved outline does not decide the answer, and a few similar-looking bumps are not enough.

Question 5

Look at the decorative shape in the figure. It has 3 similar curve marks. The marks are not placed at equal distances or equal angles. State whether the shape is symmetrical and give the reason.

?
  1. (a)It is not symmetrical because the curve marks all have the same colour.
  2. (b)It is symmetrical because it has 3 similar curve marks.
  3. (c)It is symmetrical because the decorated shape looks attractive.
  4. (d)It is not symmetrical because the similar curve marks are not repeated in a definite pattern.
Show answer

(d) It is not symmetrical because the similar curve marks are not repeated in a definite pattern.

The shape is not symmetrical. Its 3 curve marks look alike, but they are not repeated in a definite pattern. Marks of the same colour, or a pretty look, do not make a shape symmetrical.

Question 6

A group contains these objects: rangoli, pinwheel, flower, cloud, butterfly. Which object is not symmetrical in the chapter's sense?

  1. (a)flower
  2. (b)pinwheel
  3. (c)rangoli
  4. (d)cloud
Show answer

(d) cloud

The cloud is the non-example because its parts do not repeat in a definite pattern. The other listed objects are examples where parts repeat in a definite pattern.

Constructing Reflection-Symmetric Figures

Question 7

A square sheet of paper has a grid of small squares on it. It is folded along the dashed line, which runs from the bottom-left corner to the top-right corner. A hole is punched through both layers at the black dot. The dot is 8 squares from the left edge and 4 squares from the bottom edge. The sheet is then opened out. How far is the matching hole from the left edge and from the bottom edge?

012345678910012345678910fold8 right, 4 up
  1. (a)5 squares from the left edge and 8 squares from the bottom edge
  2. (b)4 squares from the left edge and 8 squares from the bottom edge
  3. (c)8 squares from the left edge and 6 squares from the bottom edge
  4. (d)2 squares from the left edge and 4 squares from the bottom edge
Show answer

(b) 4 squares from the left edge and 8 squares from the bottom edge

The fold line runs from the bottom-left corner to the top-right corner. Folding along this line swaps the two counts. The hole is 8 squares from the left edge and 4 squares from the bottom edge. So the matching hole is 4 squares from the left edge and 8 squares from the bottom edge.

Question 8

A folded sheet was punched once and then opened, leaving the two marks shown. Which line was the fold?

markmatching mark?
  1. (a)the horizontal line joining the two marks
  2. (b)the vertical line halfway between the two marks
  3. (c)the vertical line through one of the marks
  4. (d)a vertical line nearer one mark than the other
Show answer

(b) the vertical line halfway between the two marks

Matching points made by folding are mirror images. The fold line must be exactly halfway between each pair of matching points and perpendicular to the segment joining them.

Question 9

A cut was made on the folded paper shown. When the paper is unfolded, which description gives the complete pair of cut edges?

?vertical fold line
  1. (a)The unfolded paper has a second cut on the right half with the same slant and orientation as the visible cut, shifted across without reversal.
  2. (b)The unfolded paper has only the visible cut, with no matching cut on the right half.
  3. (c)The unfolded paper has a second cut on the right half with endpoints 3 and 4 square(s) from the fold.
  4. (d)The unfolded paper has the visible cut and a matching reflected cut on the right half, with endpoints 2 and 3 square(s) from the fold.
Show answer

(d) The unfolded paper has the visible cut and a matching reflected cut on the right half, with endpoints 2 and 3 square(s) from the fold.

The fold line acts as the line of symmetry. The visible cut stays where it is. A matching reflected cut appears on the other side of the fold, at the same distances from the fold.

Reflection Images and Invariants

Question 10

In the figure, point Q lies on the mirror line m. Which point is the image of Q under reflection in m?

QmTSR
  1. (a)point T, another point on m
  2. (b)point S near Q but above and left of m
  3. (c)point R below and to the right of m
  4. (d)Q itself
Show answer

(d) Q itself

A point lying on the line of symmetry does not move during reflection. Since Q is on mirror line m, the image of Q is Q itself.

Question 11

A point A is reflected in a mirror line, and its image is the point B. The point A is 7 cm away from the mirror line. A point P lies on the mirror line, away from the foot of the perpendicular from A. The length PA is 12 cm. What is the length PB?

The point and its mirror imageA and its reflected image B lie on opposite sides of the mirror line. P is on the mirror, away from the perpendicular foot. Only measurements stated in the question are labelled; PB has no length or equality mark. QUESTION DIAGRAM The point and its mirror image Schematic · measurements are supplied in the question P A B l A to mirror line: distance given in the question PA: length given in the question
  1. (a)24 cm
  2. (b)It cannot be found
  3. (c)12 cm
  4. (d)19 cm
Show answer

(c) 12 cm

B is the image of A in the mirror line, and P lies on that mirror line. So PA = PB. The distance 7 cm from A to the mirror line is not the length asked for. Since PA = 12 cm, PB = 12 cm.

Question 12

Points A and B are reflected in the mirror line l. Their images are A' and B'. The segment AB may cross line l or lie on one side of it. Write the rule that is always true about the length of AB.

ABA'B'?line l: x = 0
  1. (a)AB = AA'
  2. (b)AB is not equal to A'B'
  3. (c)AB = BB'
  4. (d)AB = A'B'
Show answer

(d) AB = A'B'

A' is the image of A and B' is the image of B. So A'B' is the image of the segment AB. Reflection changes only the position of a segment, not its length. So AB = A'B'. The segments AA' and BB' join a point to its own image, so they are not matched with AB.

Rotational Symmetry, Angles, and Order

Question 13

The figure shows a block-style capital H rotated about its marked centre. Which angle is a non-full angle of rotational symmetry for this figure?

centre?
  1. (a)90°
  2. (b)270°
  3. (c)180°
  4. (d)360°
Show answer

(c) 180°

A full turn of 360° brings every figure back to its original position. The block-style capital H also matches itself after a half-turn, so 180° is a non-full angle of rotational symmetry.

Question 14

The figure shows three equal radial arms from one centre. The angles between the arms are labelled A, B and C. Which condition makes the figure rotationally symmetric?

ABC?
  1. (a)Only A and B must be equal; C may be different.
  2. (b)Three arms are enough, whatever A, B and C are.
  3. (c)A = B = C = 60°
  4. (d)A = B = C = 120°
Show answer

(d) A = B = C = 120°

For three equal radial arms to overlap after a turn, the three intervening angles must be equal. Since angles around a point add to 360°, each angle must be 360° ÷ 3 = 120°. So A = B = C = 120°.

Question 15

The diagram marks a 45° turn about P. About which marked points does a 45° turn make the plain circle match itself?

OP45°?
  1. (a)O and P
  2. (b)Neither O nor P
  3. (c)P only
  4. (d)O only
Show answer

(d) O only

A plain circle matches after every turn about O, its centre. Turning through 45° about the off-centre point P moves its centre.

Common misconception in Symmetry

Open full misconception guide →

Does a turn guarantee a mirror line?

Wrong

A figure with rotational symmetry always has a mirror line.

Why it feels right. The shape can match itself, so any symmetry test should pass.

Right

Turns and reflections are different symmetry tests.

Worked example

The original parallelogram matches after a half turn about its centre. Reflected across the marked diagonal, the dashed outline does not match.
  1. A slanted parallelogram with unequal adjacent sides matches after a half turn.
  2. Fold along either diagonal and the sides fail to coincide.