Symmetry — Class 6 Mathematics
Ganita Prakash · Chapter 9
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- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
Chapter summary
Key ideas & terms
- Symmetry
- A property where parts of a figure repeat in a definite pattern.
- Line of symmetry
- A line that divides a figure into two parts that exactly overlap when folded along it; also called an axis of symmetry.
- Reflection symmetry
- The symmetry a figure has when it possesses one or more lines of symmetry.
- Rotational symmetry
- The property of looking exactly the same after a rotation of less than
about a fixed point. - Angle of symmetry
- An angle through which a figure can be rotated to look exactly the same.
- Centre of rotation
- The fixed point about which a figure is rotated.
Extra questions & answers
What is a line of symmetry?
A line that cuts a figure into two parts that exactly overlap when the figure is folded along that line.
How many lines of symmetry does a square have?
A square has 4 lines of symmetry \u2014 two through the midpoints of opposite sides and two along the diagonals.
Is a diagonal of a non-square rectangle a line of symmetry?
No; folding a rectangle (that is not a square) along its diagonal does not make the halves overlap.
How many lines of symmetry does a circle have?
Infinitely many; every diameter of a circle is a line of symmetry.
What is reflection symmetry?
A figure has reflection symmetry if it has at least one line of symmetry, so one side is the mirror image of the other.
What is rotational symmetry?
A figure has rotational symmetry if it looks exactly the same after a rotation of less than
What is an angle of symmetry?
An angle through which a figure can be rotated so that it looks exactly the same;
What are the angles of symmetry of a figure whose smallest angle of symmetry is
If the smallest angle of symmetry is a whole number of degrees, what must it divide?
It must be a factor of
Can a figure have angles of symmetry but no line of symmetry?
Yes; for example a windmill has rotational symmetry but no line of symmetry.
Figure it Out
1. Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
The flower has
2. For each of the following figures, identify the line(s) of symmetry if it exists.
From left to right: the notched arrow has 1 line of symmetry (along its length); the kite has 1 line of symmetry (its main axis); the house/trapezium shape has 0 lines of symmetry; the L-shaped figure has 1 line of symmetry (its diagonal); and the triangle shown has 0 lines of symmetry.
Figure it Out
1. In each of the following figures, a hole was punched in a folded square sheet of paper and then 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?
a. The two holes are mirror images left-right, so the fold was a vertical line through the centre. b. The two holes are diagonal mirror images, so the fold was a diagonal from top left to bottom right. c. The two holes are mirror images top-bottom, so the fold was a horizontal line through the centre. d. The four holes are symmetric both vertically and horizontally, so a single punch produced four holes. The paper was folded twice - once vertically and once horizontally (in either order) and the punch was made through all four layers.
2. Given the line(s) of symmetry, find the other hole(s):
The missing hole is the mirror image of the given hole across the dotted line(s) of symmetry. a. Reflect across the diagonal to get a hole in the lower right region. b. Reflect across the horizontal mid-line to get a hole in the upper half. c. Reflect across the vertical altitude to get a hole symmetric on the left side. d. and e. Reflect the given hole through the centre across the dotted diameter to get the diametrically opposite hole inside the circle.
4. After each of the following cuts, predict the shape of the hole when the paper is opened. After you have made your prediction, make the cutouts and verify your answer.
a. A vertical fold with a zig-zag cut on the folded edge opens to a symmetric diamond-like hexagon with pointed left and right vertices. b. A horizontal fold with a triangular notch cut from the fold opens to a symmetric diamond hole in the middle of the sheet. c. The blue paper is folded vertically then horizontally and two small rectangular notches are cut from the folded edges; when opened it has a small central rectangular hole and two open notches at the middle of the top and bottom edges. d. The yellow paper is folded vertically and a rectangular strip is cut from the middle of the folded edge; when opened it forms a single central rectangular hole that connects the left and right halves, looking like an
5. Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it? (a) The hole in the centre is a square.
Fold the square sheet in half vertically and then horizontally so that the centre of the sheet becomes the closed corner where all folds meet. Keeping the folds closed, cut a small square with sides parallel to the edges of the sheet, with one corner at the closed centre. On unfolding you get a central square hole whose sides are parallel to the outer border.
5. Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it? (b) The hole in the centre is a square.
Fold the sheet in the same way, vertically and horizontally to bring the centre to the closed corner. Now make a single straight slanting cut at
6. How many lines of symmetry do these shapes have? (a)
The star-like octagon with alternating pointed and flat vertices has
6. How many lines of symmetry do these shapes have? (b) A triangle with equal sides and equal angles.
An equilateral triangle has
6. How many lines of symmetry do these shapes have? (c) A hexagon with equal sides and equal angles.
A regular hexagon has
7. Trace each figure and draw the lines of symmetry, if any:
Draw the figures and test by folding. The symmetrical figures among them have one or more mirror lines through their centres. For example a circle would have infinite lines, an isosceles triangle one, a regular polygon as many as its sides. Any irregular scalene shape in the set has no line of symmetry.
8. Find the lines of symmetry for the kolam below.
The kolam has 6 lines of symmetry (it is based on a hexagonal design, so it has six axes of reflection).
9. Draw the following. (a) A triangle with exactly one line of symmetry.
An isosceles triangle with two equal sides and base angles equal, but not equilateral, has exactly one line of symmetry which is the altitude from the apex to the base.
9. Draw the following. (b) A triangle with exactly three lines of symmetry.
An equilateral triangle with all three sides equal and all angles
9. Draw the following. (c) A triangle with no line of symmetry.
A scalene triangle with all sides of different lengths and all angles different has no line of symmetry.
9. Is it possible to draw a triangle with exactly two lines of symmetry?
No. A triangle cannot have exactly two lines of symmetry. If it has two, the third automatically exists and it becomes equilateral with three. So the possible numbers are
10. Draw the following. In each case, the figure should contain at least one curved boundary. (a) A figure with exactly one line of symmetry.
A semicircle or a heart shape or the letter
10. Draw the following. In each case, the figure should contain at least one curved boundary. (b) A figure with exactly two lines of symmetry.
An ellipse with distinct major and minor axes, or a shape made of two overlapping circles like a figure-of-eight oriented symmetrically, has exactly two perpendicular lines of symmetry.
10. Draw the following. In each case, the figure should contain at least one curved boundary. (c) A figure with exactly four lines of symmetry.
A circle with four equally spaced petals or a shape formed by a square with semicircles on each side has four lines of symmetry, like a square.
11. Copy the following on squared paper. Complete them so that the blue line is a line of symmetry. Problem (a) has been done for you.
Complete each half by reflecting every square and dot across the blue mirror line so that the distance from the line is preserved. For (c) and (f) rotate the page by
12. Copy the following drawing on squared paper. Complete each one of them so that the resulting figure has the two blue lines as lines of symmetry.
Reflect the given part first across one blue line and then across the other. The final figure will be four times the original quadrant and symmetric about both the vertical and horizontal blue lines through the centre.
13. Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
Add two segments so the new polygon becomes isosceles or kite-like. For example join the open ends to form a symmetric pentagon or an arrow where the added lines are mirror images across the intended line of symmetry.
Figure it Out
1. Find the angles of symmetry for the given figures about the point marked •. (a)
Figure (a) is a four-armed windmill with equal arms at
1. Find the angles of symmetry for the given figures about the point marked •. (b)
Figure (b) has no rotational symmetry except the full turn. Its only angle of symmetry is
1. Find the angles of symmetry for the given figures about the point marked •. (c)
Figure (c) has two-fold symmetry. Its angles of symmetry are
2. Which of the following figures have more than one angle of symmetry?
Figures that look the same after a rotation smaller than
3. Give the order of rotational symmetry for each figure:
The order of rotational symmetry is the number of times a figure looks exactly the same during one full turn. For the six given figures the orders are: (a) 2, (b) 4, (c) 6, (d) 3, (e) 4, (f) 5.
True or False: Every figure will have 360 degrees as an angle of symmetry.
True. Rotating any figure by
True or False: If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.
True. The smallest angle
Figure it Out
1. Colour the sectors of the circle below so that the figure has i) 3 angles of symmetry,
Divide the circle into
1. Colour the sectors of the circle below so that the figure has ii) 4 angles of symmetry,
Colour opposite sectors alike to get four-fold symmetry. For example colour alternate sectors with two colours repeating every
1. Colour the sectors of the circle below so that the figure has iii) what are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?
With
2. Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
An equilateral triangle has
3. Draw, wherever possible, a rough sketch of: (a) A triangle with at least two lines of symmetry and at least two angles of symmetry.
An equilateral triangle has
3. Draw, wherever possible, a rough sketch of: (b) A triangle with only one line of symmetry but not having rotational symmetry.
An isosceles triangle that is not equilateral has exactly one line of symmetry and only
3. Draw, wherever possible, a rough sketch of: (c) A quadrilateral with rotational symmetry but no reflection symmetry.
A parallelogram that is not a rhombus or rectangle, like a slanted parallelogram with opposite sides parallel but adjacent sides unequal and angles not
3. Draw, wherever possible, a rough sketch of: (d) A quadrilateral with reflection symmetry but not having rotational symmetry.
An isosceles trapezium with one pair of parallel sides and non-parallel sides equal has one line of symmetry through the midpoints of the parallel sides but only
4. In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
Multiples of
5. 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?
If
6. Can we have a figure with rotational symmetry whose smallest angle of symmetry is: (a) 45°?
Yes.
6. Can we have a figure with rotational symmetry whose smallest angle of symmetry is: (b) 17°?
No.
7. This is a picture of the new Parliament Building in Delhi. (a) Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetries. How many are they?
Yes, the outer boundary is an equilateral triangle with rounded corners and has
7. This is a picture of the new Parliament Building in Delhi. (b) Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.
Yes, it has rotational symmetry of order
8. How many lines of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?
Triangle
9. How many angles of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?
The number of angles of rotational symmetry equals the number of sides. So triangle has
10. How many lines of symmetry do the shapes in the last shape sequence in Chapter 1, Table 3, the Koch Snowflake sequence, have? How many angles of symmetry?
Each Koch snowflake in the sequence is based on an equilateral triangle and retains
11. How many lines of symmetry and angles of symmetry does Ashoka Chakra have?
The Ashoka Chakra has