Welcome dear students! Today we are going to learn about Symmetry from Class 6 Maths. Look around you. You may find many objects that catch your attention, like a flower, a butterfly, a rangoli, and a pinwheel. There is something beautiful about these pictures. The flower looks the same from many different angles. What about the butterfly? The colours are very attractive, but what else appeals to you? In these pictures, it appears that some parts of the figure are repeated and these repetitions occur in a definite pattern. Can you see what repeats in the beautiful rangoli figure? In the rangoli, the red petals come back onto themselves when the flower is rotated by 90° around the centre. What about the pinwheel? Look at the hexagon first. Can you say what figure repeats along each side? How do these shapes move as you move along the boundary? On the other hand, look at a picture of clouds. There is no such repetitive pattern. We can say that the first four figures are symmetrical and the cloud is not symmetrical. A symmetry refers to a part or parts of a figure that are repeated in some definite pattern. What are the symmetries that you see in the Taj Mahal and the Gopuram?
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Let us move to section 9.1, Line of Symmetry. Figure a shows a blue triangle with a dotted line. What if you fold the triangle along the dotted line? One half of the triangle covers the other half completely. These are called mirror halves! What about Figure b with four puzzle pieces and a dotted line passing through the middle? Are they mirror halves? No, when we fold along the line, the left half does not exactly fit over the right half. A line that cuts a figure into two parts that exactly overlap when folded along that line is called a line of symmetry of the figure. Now let us do the first Figure it Out. Do you see any line of symmetry in the figures at the start of the chapter? The flower, butterfly, rangoli, and pinwheel all have lines of symmetry. What about the cloud? No, the cloud has no line of symmetry. For the second question, the textbook shows several figures and asks you to identify their lines of symmetry if they exist. Look at each figure carefully. If you can fold it so the halves match perfectly, that fold line is a line of symmetry. Some figures may have one, some may have more than one, and some may have none.
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Now let us explore figures with more than one line of symmetry. Does a square have only one line of symmetry? Take a square piece of paper. By folding, find all its lines of symmetry. Fold one: fold the paper in half vertically. Fold two: fold it again in half horizontally. Now open out the folds. You see a vertical fold and a horizontal fold. Fold three: fold the square in half along a diagonal. Open it. Fold four: fold it in half along the other diagonal. Open it. Is there any other way to fold the square so that the two halves overlap? No. How many lines of symmetry does the square have? Four. Thus, figures can have multiple lines of symmetry. We saw that the diagonal of a square is also a line of symmetry. Let us take a rectangle that is not a square. Is its diagonal a line of symmetry? Take a rectangular piece of paper and check if the two parts overlap by folding it along its diagonal. You will observe that the two parts do not overlap exactly. So the diagonal of a non-square rectangle is not a line of symmetry.
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So far we have said that when we fold a figure along a line of symmetry, the two parts overlap completely. We could also say that the part of the figure on one side of the line of symmetry is reflected by the line to the other side. Let us understand this by labeling points. The figure shows a square with corners labeled A, B, C and D. Consider the vertical line of symmetry. When we reflect the square along this line, points B and C on the right get reflected to the left side and occupy the positions of A and D. What happens to points A and D? A occupies the position of B, and D occupies the position of C. What if we reflect along the diagonal from A to C? Points A and C stay in place. Point B moves to where D was, and D moves to where B was. What if we reflect along the horizontal line of symmetry? Points A and B swap places, and D and C swap places. A figure that has a line or lines of symmetry is said to have reflection symmetry.
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Now let us learn how to generate shapes having lines of symmetry. Ink Blot Devils: Take a piece of paper. Fold it in half. Open it and spill a few drops of ink or paint on one half. Press the halves together and open again. What do you see? A symmetric ink pattern. Is it symmetric? Yes. Where is the line of symmetry? The fold line. Is there any other line? Usually no. Try making more such patterns. Paper Folding and Cutting: Here is another way! In two figures, a sheet is folded and a cut is made along a dotted line. Draw how it looks when unfolded. The cut will appear mirrored across the fold line, creating a symmetric shape. The fold line is the line of symmetry. Make different symmetric shapes by folding and cutting. Use thin rectangular coloured paper. Fold it several times and create intricate patterns by cutting. Identify the lines of symmetry in the repeating design. Use such decorative cut outs for festive occasions.
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Let us work through the Figure it Out Punching Game exercises. The fold is a line of symmetry. Punch holes at different locations of a folded square sheet and create symmetric patterns. Exercise 1: In each figure, a hole was punched in a folded square and then unfolded. Identify the fold line. For figure a, a square with two holes on left and right sides at middle height, the paper was folded along the vertical midline. For figure b, a square with two holes in the top right corner, the paper was folded along the diagonal from top left to bottom right. For figure c, a square with two holes on the right side, top and bottom, the paper was folded along the horizontal midline. For figure d, a square with four holes in all four corners, the paper was folded twice, first vertically then horizontally, or along both diagonals.
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Exercise 2: Given the line of symmetry, find the other hole. For a, a square with a diagonal dashed line from top left to bottom right and one hole in the top left corner, the reflected hole goes to the bottom right corner. For b, a square with a horizontal dashed line through the middle and one hole in the bottom right, the reflected hole goes to the top right corner. For c, a triangle with a vertical dashed line from top vertex to base and one hole in the top portion on the right, the reflected hole goes to the top portion on the left. For d, a circle with a diagonal dashed line and one hole in the upper right, the reflected hole goes to the lower left. For e, a circle with a diagonal dashed line and one hole in the upper left, the reflected hole goes to the lower right.
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Exercise 3 introduces how we represent paper folds. A vertical fold is shown with a diagram of paper folded vertically with an arrow indicating the direction. A horizontal fold is represented similarly. Exercise 4 asks you to look at four different folded paper diagrams, each with a cut along a dotted line. For each cut, predict the shape of the hole when the paper is opened. The textbook encourages you to make your prediction first, and then actually fold a piece of paper, make the cut, and open it to verify your answer. This hands-on activity helps you see how the fold line acts as a mirror for the cut. Exercise 5 challenges you to think in reverse. You are shown two target shapes, each with a square hole in the centre. The question asks how you would use folds and a single straight cut to create these shapes. Try folding your paper to create the necessary lines of symmetry, make one straight cut, and then open it to see if you get the square hole. Remember to check if the resulting four-sided figure satisfies the properties of a square.
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Exercise 6 asks how many lines of symmetry certain shapes have. For the first shape, which is a scalene triangle, there are zero lines of symmetry. For the second, a triangle with equal sides and equal angles, which is an equilateral triangle, there are three lines of symmetry. For the third, a hexagon with equal sides and equal angles, which is a regular hexagon, there are six lines of symmetry. Exercise 7 asks you to trace each figure and draw the lines of symmetry. The isosceles triangle has one vertical line. The rectangle has two lines. The regular pentagon has five lines. The circle has infinite lines. Exercise 8 asks you to find the lines of symmetry for the kolam. The kolam has a hexagonal pattern with six-fold symmetry. It has six lines of symmetry passing through opposite vertices and midpoints of opposite sides. Exercise 9: Draw a triangle with exactly one line of symmetry: draw an isosceles triangle. A triangle with exactly three lines: draw an equilateral triangle. A triangle with no line of symmetry: draw a scalene triangle. Is it possible to draw a triangle with exactly two lines? No, it is not possible.
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Exercise 10: Draw figures with at least one curved boundary. Exactly one line of symmetry: draw a semicircle. Exactly two lines: draw an ellipse. Exactly four lines: draw a circle with a square inscribed, or a four-petal flower. Exercise 11: Copy on squared paper and complete so the blue line is a line of symmetry. For diagram b, reflect each red square across the horizontal blue line. For diagram c, reflect across the diagonal blue line from bottom left to top right. For d, reflect across the vertical line. For e, reflect across the horizontal line. For f, reflect across the diagonal line. Hint: For c and f, rotating the book helps! Exercise 12: Copy and complete so the figure has two blue lines as lines of symmetry. Reflect the given red squares across both vertical and horizontal blue lines to create a four-quadrant symmetric pattern. Exercise 13: Copy on a dot grid. For each figure, draw two more lines to make a shape that has a line of symmetry. Connect the dots to form symmetric polygons like isosceles triangles or kites.
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Let us move to section 9.2, Rotational Symmetry. The paper windmill looks symmetrical but has no line of symmetry! If you fold it, the halves will not overlap. But if you rotate it by 90° about the red point at the centre, it looks exactly the same. We say the windmill has rotational symmetry. There is always a fixed point about which the object is rotated. This is called the centre of rotation. Will it look the same when rotated less than 90°? No! An angle through which a figure can be rotated to look exactly the same is called an angle of rotational symmetry. For the windmill, the angles of symmetry are 90°, 180°, 270° and 360°. When any figure is rotated by 360°, it returns to its original position, so 360° is always an angle of symmetry. The windmill has 4 angles of symmetry.
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What other shape has exactly four angles of symmetry? A square. It requires 90° rotation to overlap with itself. This takes point A to B, B to C, C to D, and D to A. The centre of rotation is at the intersection of the diagonals. The other angles are 180°, 270°, and 360°. Now, a worked example: Find the angles of symmetry of a trapezoid strip with a dot at its centre. Rotate it clockwise. A 180° rotation results in an inverted trapezoid. Does it overlap? No, because the longer base moves to where the shorter base was. Another 180° rotation gives the original shape. It only matches after a full 360° turn. So, this figure does not have rotational symmetry.
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Now consider Rotational Symmetry of Figures with Radial Arms. A figure with 4 radial arms has angles of symmetry at 90°, 180°, 270°, and 360°. The angle between adjacent arms is 90°. To get only two angles of symmetry, make two opposite arms longer and the other two shorter. Can we get exactly 3 angles? Try with 3 radial arms. Only 360° brings it back. But if we change the angles between arms to be equal, it works. For overlap, angle A must equal angle B must equal angle C. A full turn is 360°, so each angle is 360°/3 = 120°. The angles of rotation are 120°, 240°, and 360°. For 5 arms, the angle is 360°/5 = 72°. Angles: 72°, 144°, 216°, 288°, 360°. For 6 arms, 360°/6 = 60°. For 7 arms, 360°/7 = 51 and 3/7 degrees.
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Figure it Out 1: Find angles of symmetry about the marked point. For a plus sign, angles are 90°, 180°, 270°, 360°. For an L-shape, only 360°. For a T-shape, only 360°. Figure it Out 2: Which have more than one angle? The circle with perpendicular diameters has 4. Equilateral triangle has 3. Circle with 3 radial lines has 3. Four-leaf clover has 4. X shape has 2. Five-pointed star has 5. B shape has only 360°. All except the B shape have more than one. Figure it Out 3: Give the order of rotational symmetry. Line segment with arrows: 2. X shape: 2. Six-pointed star: 6. Running human figure: 1. Cross of five squares: 4. Regular pentagon: 5.
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Let us list the angles. For 2 angles: 180°, 360°. For 3 angles: 120°, 240°, 360°. For 4 angles: 90°, 180°, 270°, 360°. The angles are all multiples of the smallest angle. True or False: Every figure will have 360° as an angle of symmetry. True. If the smallest angle is a natural number, it is a factor of 360. True. Is there a smallest angle for all figures? Most, except the circle. Symmetries of a circle: Rotating a circle about its centre makes it coincide with itself at any angle. Every angle is an angle of symmetry. Every diameter is a line of symmetry. Other objects: Fan, Flower, Wheel.
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Next Figure it Out. Exercise 1: Colour sectors to get 3 angles of symmetry by colouring every third sector. For 4 angles, colour every fourth sector. Possible numbers are factors of the total sectors. Exercise 2: Draw two figures other than a circle and square with both reflection and rotational symmetry: an equilateral triangle and a regular hexagon. Exercise 3: Draw sketches. Triangle with at least two lines and angles: equilateral. Triangle with one line but no rotational symmetry: isosceles. Quadrilateral with rotational but no reflection symmetry: parallelogram. Quadrilateral with reflection but no rotational symmetry: kite. Exercise 4: Smallest angle 60°. Others: 120°, 180°, 240°, 300°, 360°. Exercise 5: In a figure, 60° is an angle of symmetry and there are exactly two angles of symmetry smaller than 60°. The angles of symmetry must be multiples of the smallest angle. Since 60° is the third multiple, we divide 60° by 3 to find the smallest angle. The smallest angle of symmetry is 20°. The full set of angles less than 360° would be 20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°, 180°, 200°, 220°, 240°, 260°, 280°, 300°, 320°, 340°, and 360°. Exercise 6: Can smallest be 45°? Yes. 17°? No.
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Exercise 7: For the picture of the new Parliament Building in Delhi, the outer boundary has multiple lines of reflection symmetry. It also has rotational symmetry around its centre. The exact number of lines and angles depends on the specific polygonal design shown in the textbook diagram, but regular symmetrical structures like this possess both reflection and rotational symmetry. Exercise 8: For the first shape sequence of regular polygons from Chapter 1, which are the equilateral triangle, square, regular pentagon, and regular hexagon, the number of lines of symmetry are 3, 4, 5, and 6 respectively. This gives the number sequence 3, 4, 5, 6. Exercise 9: For the same regular polygons, the number of angles of symmetry equals the number of sides. So the equilateral triangle has 3, the square has 4, the regular pentagon has 5, and the regular hexagon has 6. The sequence is again 3, 4, 5, 6. Exercise 10: For the Koch Snowflake sequence, the limiting fractal shape has 6 lines of symmetry and 6 angles of symmetry. Please note that this refers to the final limiting shape, not the intermediate construction stages. Exercise 11: Ashoka Chakra has 24 lines and 24 angles of symmetry. Playing with Tiles: Complete a figure with exactly 2 lines by placing tiles symmetrically across both axes. Use 16 tiles for 1 line by arranging in a symmetric row. Use 16 for 2 lines by making a square or cross. Game: Draw a 6 by 6 grid. Players cover two adjacent squares. The player unable to place a line loses. Strategy: Mirror your opponent's moves across the centre to maintain symmetry and control the board.
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Let us review the Summary. When a figure is made up of parts that repeat in a definite pattern, we say that the figure has symmetry. We say that such a figure is symmetrical. A line that cuts a plane figure into two parts that exactly overlap when folded along that line is called a line of symmetry or axis of symmetry of the figure. A figure may have multiple lines of symmetry. Sometimes a figure looks exactly the same when it is rotated by an angle about a fixed point. Such an angle is called an angle of symmetry of the figure. A figure that has an angle of symmetry strictly between 0 and 360 degrees is said to have rotational symmetry. The point of the figure about which the rotation occurs is called the centre of rotation. A figure may have multiple angles of symmetry. Some figures may have a line of symmetry but no angle of symmetry, while others may have angles of symmetry but no lines of symmetry. Some figures may have both lines of symmetry as well as angles of symmetry.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]