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09
NCERT Solutions · Maths Ch 9
Symmetry
42 questions · 4 exercises
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Ex 9.1
Ex 9.2
Ex 9.3
Ex 9.4
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 9.1 · 7 QUESTIONS
Lines of Symmetry
Q1
Do you see any line of symmetry in the figures at the start of the chapter (flower, rangoli, butterfly, pinwheel)? What about in the picture of the cloud?
(2 marks)
Q2
For each of the following figures, identify the line(s) of symmetry if it exists.
(2 marks)
Q3
Does a square have only one line of symmetry? Find all its lines of symmetry by folding. Is there any other way to fold the square so that the two halves overlap?
(3 marks)
Q4
Take a rectangle that is not a square. Is its diagonal a line of symmetry?
(2 marks)
Q5
Reflection: for the square ABCD (A top-left, B top-right, C bottom-right, D bottom-left), where do A, B, C and D go on reflection (a) in the vertical line of symmetry, (b) along the diagonal AC, (c) in the horizontal line of symmetry?
(2 marks)
Q6
Ink Blot Devils: fold a paper in half, spill ink on one half, press and open. Is the resulting figure symmetric? Where is the line of symmetry?
(2 marks)
Q7
Paper folding and cutting: a sheet of paper is folded and a cut is made along the dotted line. Draw a sketch of how the paper will look when unfolded. Do you see a line of symmetry?
(2 marks)
2
EXERCISE 9.2 · 12 QUESTIONS
Punching, Cutting and Completing Figures
Q1
Punching Game (Q1): in each figure (a to d) a hole was punched in a folded square sheet and the paper was unfolded. Identify the line along which the paper was folded. Figure (d) was created by punching a single hole: how was the paper folded?
(2 marks)
Q2
Given the line(s) of symmetry, find the other hole(s) (figures a to e).
(1 mark)
Q3
Paper cutting: after each of the cuts shown (a to d), predict the shape of the hole when the paper is opened.
(2 marks)
Q4
Suppose you have to get each of these shapes with some folds and a single straight cut: (a) the hole in the centre is a square; (b) the hole in the centre is a square (with its sides slanting).
(2 marks)
Q5
How many lines of symmetry do these shapes have? (b) A triangle with equal sides and equal angles; (c) A hexagon with equal sides and equal angles.
(2 marks)
Q6
Trace each figure and draw the lines of symmetry, if any. (Q7)
(2 marks)
Q7
Find the lines of symmetry for the kolam below.
(1 mark)
Q8
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?
(3 marks)
Q9
Draw the following; in each case the figure should contain at least one curved boundary: (a) exactly one line of symmetry (b) exactly two lines of symmetry (c) exactly four lines of symmetry.
(1 mark)
Q10
Copy the figures on squared paper and complete them so that the blue line is a line of symmetry (a to f). Hint: for (c) and (f), see if rotating the book helps.
(1 mark)
Q11
Complete each figure so that the resulting figure has the two blue lines as lines of symmetry (a to f).
(1 mark)
Q12
Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
(1 mark)
3
EXERCISE 9.3 · 11 QUESTIONS
Rotational Symmetry
Q1
The windmill has rotational symmetry. What are its angles of symmetry? How many?
(2 marks)
Q2
How many angles of symmetry does a square have? What are they, and where is the centre of rotation?
(2 marks)
Q3
Find the angles of symmetry of the strip drawn in the book (an arrow-like strip).
(2 marks)
Q4
A picture has 4 radial arms with 90° between adjacent arms. How many angles of symmetry does it have? How can it be changed to have only two angles of symmetry?
(2 marks)
Q5
A figure has 3 radial arms. For what angles between adjacent arms does it have 3 angles of symmetry? What are the angles?
(2 marks)
Q6
Can you draw a figure with radial arms that has (a) exactly 5 angles of symmetry (b) 6 angles of symmetry? Find the angles in each case.
(2 marks)
Q7
A figure with radial arms has exactly 7 angles of symmetry. What is its smallest angle of symmetry? Is it a whole number? If not, express it as a mixed fraction.
(2 marks)
Q8
Find the angles of symmetry for the given figures about the marked point: (a), (b), (c).
(2 marks)
Q9
Which of the given figures have more than one angle of symmetry? Give the order of rotational symmetry of each figure.
(2 marks)
Q10
Do the angles of symmetry always come in multiples of the smallest angle? True or False: (i) Every figure has 360° as an angle of symmetry. (ii) If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.
(2 marks)
Q11
Symmetries of a circle: how many angles of symmetry does a circle have? How many lines of symmetry?
(2 marks)
4
12 QUESTIONS
Exercise 9.4
Q1
Colour the sectors of a circle (divided into equal sectors) so that the figure has (i) 3 angles of symmetry, (ii) 4 angles of symmetry, (iii) what are the possible numbers of angles of symmetry you can obtain by colouring in different ways?
(2 marks)
Q2
Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
(2 marks)
Q3
Draw, wherever possible, a rough sketch of: (a) a triangle with at least two lines and at least two angles of symmetry; (b) a triangle with only one line of symmetry but no rotational symmetry; (c) a quadrilateral with rotational symmetry but no reflection symmetry; (d) a quadrilateral with reflection symmetry but no rotational symmetry.
(2 marks)
Q4
In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
(2 marks)
Q5
In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?
(2 marks)
Q6
Can we have a figure with rotational symmetry whose smallest angle of symmetry is (a) 45°? (b) 17°?
(1 mark)
Q7
The new Parliament Building in Delhi: (a) Does the outer boundary of the picture have reflection symmetry? How many lines? (b) Does it have rotational symmetry about its centre? Find the angles.
(2 marks)
Q8
How many lines of symmetry do the regular polygons (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon) of the first shape sequence of Chapter 1 have? What number sequence do you get?
(3 marks)
Q9
How many angles of symmetry do the same regular polygons have? What number sequence do you get?
(2 marks)
Q10
How many lines of symmetry and angles of symmetry do the shapes in the last shape sequence of Chapter 1, Table 3 (the Koch Snowflake sequence), have?
(2 marks)
Q11
How many lines of symmetry and angles of symmetry does the Ashoka Chakra have?
(2 marks)
Q12
Game: draw a 6 by 6 grid. Two players take turns covering two adjacent squares by drawing a line (horizontal or vertical, lines cannot overlap). The player who cannot place a line loses. With what strategy can one win?
(2 marks)