Welcome dear students! Today we are going to learn about Symmetry from Class 7 Maths. Symmetry is an important geometrical concept, commonly exhibited in nature and is used almost in every field of activity. Artists, professionals, designers of clothing or jewellery, car manufacturers, architects and many others make use of the idea of symmetry. The beehives, the flowers, the tree-leaves, religious symbols, rugs, and handkerchiefs — everywhere you find symmetrical designs. You have already had a feel of line symmetry in your previous class. A figure has a line symmetry, if there is a line about which the figure may be folded so that the two parts of the figure will coincide. You might like to recall these ideas. Here are some activities to help you. You can create some colourful ink-dot devils, make some symmetrical paper-cut designs, or compose a picture-album showing symmetry.
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Enjoy identifying lines, also called axes, of symmetry in the designs you collect. Let us now strengthen our ideas on symmetry further. Study the following figures in which the lines of symmetry are marked with dotted lines. Figure 12.1 shows four different shapes, each with a dotted line running through the center, dividing them into matching halves. Now we move to section 12.2, Lines of Symmetry for Regular Polygons. You know that a polygon is a closed figure made of several line segments. The polygon made up of the least number of line segments is the triangle. Can there be a polygon that you can draw with still fewer line segments? Think about it. A polygon is said to be regular if all its sides are of equal length and all its angles are of equal measure. Thus, an equilateral triangle is a regular polygon of three sides. Can you name the regular polygon of four sides? An equilateral triangle is regular because each of its sides has same length and each of its angles measures 60°. In Figure 12.2, we see a triangle with all three sides marked equal and each angle labeled 60°.
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A square is also regular because all its sides are of equal length and each of its angles is a right angle, that is, 90°. Its diagonals are seen to be perpendicular bisectors of one another, as shown in Figure 12.3. If a pentagon is regular, naturally, its sides should have equal length. You will, later on, learn that the measure of each of its angles is 108°, as shown in Figure 12.4. A regular hexagon has all its sides equal and each of its angles measures 120°. You will learn more of these figures later, as seen in Figure 12.5. The regular polygons are symmetrical figures and hence their lines of symmetry are quite interesting. Each regular polygon has as many lines of symmetry as it has sides, shown in Figure 12.6. We say, they have multiple lines of symmetry. Perhaps, you might like to investigate this by paper folding. Go ahead! The concept of line symmetry is closely related to mirror reflection. A shape has line symmetry when one half of it is the mirror image of the other half, as shown in Figure 12.7. A mirror line, thus, helps to visualise a line of symmetry, as shown in Figure 12.8. In this figure, we check if a dotted line is a mirror line. For the first shape, the answer is no. For the second shape, the answer is yes.
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While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation, as seen in Figure 12.9. The shape is same, but the other way round! Now, let us play this punching game. Fold a sheet into two halves. Punch a hole or two holes about the symmetric fold. The fold is a line, or axis, of symmetry. Study about punches at different locations on the folded paper and the corresponding lines of symmetry, as illustrated in Figure 12.10. Now it is time to practice. Let us work through Exercise 12.1 together. Question 1 asks you to copy the figures with punched holes and find the axes of symmetry. The first figure shows two holes aligned horizontally. To find the axis of symmetry, draw a vertical line exactly halfway between them. Folding along this line makes the holes coincide. The second figure shows two holes aligned vertically. Draw a horizontal line exactly halfway between them. Folding along this line makes the holes coincide. Question 2 asks you to find the other hole given the line of symmetry. Locate the given hole, measure its perpendicular distance to the given line of symmetry, and mark a new point on the opposite side of the line at the exact same perpendicular distance. This new point is the position of the other hole.
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Question 3 gives you a mirror line as a dotted line and asks you to complete each figure by performing reflection. To solve this, take each vertex or corner of the given half. Measure its perpendicular distance to the dotted mirror line. Plot the reflected point at the same distance on the opposite side. Connect these new points in the same order as the original. The completed figure will be a full symmetrical shape, such as a square, rectangle, or rhombus, depending on the original half. Question 4 asks you to identify multiple lines of symmetry in various figures. For a square, you will find four lines: two diagonals and two midlines. For a rectangle, you will find two lines: the vertical and horizontal midlines. For a circle, you will find infinite lines passing through the center. For a regular hexagon, you will find six lines: three through opposite vertices and three through midpoints of opposite sides. Question 5 asks you to take a diagonal as a line of symmetry and shade more squares to make the figure symmetric. Identify the diagonal as the mirror line. For every shaded square on one side, shade the corresponding square on the other side at the same perpendicular distance from the diagonal. Yes, there can be more than one way depending on your initial choices. If you shade squares symmetrically with respect to both diagonals, the figure will be symmetric about both.
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Question 6 asks you to complete each shape to be symmetric about the mirror line. Follow the same reflection steps as in question three. Measure the distance of each point from the mirror line, plot the mirror image point, and connect them to complete the shape. Question 7 asks for the number of lines of symmetry for several figures. Let us solve each one step by step. An equilateral triangle has three lines of symmetry, one from each vertex to the midpoint of the opposite side. An isosceles triangle has one line of symmetry, from the vertex angle to the midpoint of the base. A scalene triangle has zero lines of symmetry, as all sides and angles are different. A square has four lines of symmetry. A rectangle has two lines of symmetry. A rhombus has two lines of symmetry, along its diagonals. A parallelogram has zero lines of symmetry. A general quadrilateral has zero lines of symmetry. A regular hexagon has six lines of symmetry. A circle has infinite lines of symmetry. Question 8 asks which English letters have reflectional symmetry. For a vertical mirror, the letters are A, H, I, M, O, T, U, V, W, X, Y. For a horizontal mirror, the letters are B, C, D, E, H, I, K, O, X. For both horizontal and vertical mirrors, the letters are H, I, O, X.
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Question 9 asks for three examples of shapes with no line of symmetry. Examples include a scalene triangle, a general parallelogram, and an irregular quadrilateral. Question 10 asks what other name you can give to the line of symmetry of an isosceles triangle and a circle. For an isosceles triangle, the line of symmetry is also called the median, altitude, or angle bisector from the vertex angle. For a circle, any line of symmetry is a diameter. Now we move to section 12.3, Rotational Symmetry. What do you say when the hands of a clock go round? You say that they rotate. The hands of a clock rotate in only one direction, about a fixed point, the centre of the clock-face. Rotation, like movement of the hands of a clock, is called a clockwise rotation; otherwise it is said to be anticlockwise. What can you say about the rotation of the blades of a ceiling fan? Do they rotate clockwise or anticlockwise? Or do they rotate both ways? If you spin the wheel of a bicycle, it rotates in either way: both clockwise and anticlockwise. Give three examples each for clockwise and anticlockwise rotation. For clockwise, think of a clock, a screw being tightened, or a ceiling fan on standard mode. For anticlockwise, think of a screw being loosened, a ceiling fan in reverse mode, or a steering wheel turning left.
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When an object rotates, its shape and size do not change. The rotation turns an object about a fixed point. This fixed point is the centre of rotation. What is the centre of rotation of the hands of a clock? It is the central pivot point where the hands meet. The angle of turning during rotation is called the angle of rotation. A full turn means a rotation of 360°. What is the degree measure for a half-turn? It is 180°. What about a quarter-turn? It is 90°. When it is 12 O’clock, the hands are together. By 3 O’clock, the minute hand has made three complete turns, but the hour hand has made only a quarter-turn. At 6 O’clock, the hands point in exactly opposite directions, forming a straight line. Have you ever made a paper windmill? The paper windmill in Figure 12.11 looks symmetrical, but you do not find any line of symmetry. No folding can give you coincident halves. However, if you rotate it by 90° about the fixed point, the windmill looks exactly the same. We say the windmill has rotational symmetry. In a full turn, there are precisely four positions, at 90°, 180°, 270°, and 360°, when the windmill looks exactly the same. Because of this, we say it has a rotational symmetry of order 4.
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Let us look at one more example for rotational symmetry. Consider a square with P as one of its corners, shown in Figure 12.13. Let us perform quarter-turns about the centre of the square. The first image is the initial position. Rotation by 90° about the centre leads to the second position. Note the position of P now. Rotate again through 90° and you get the third position. When you complete four quarter-turns, the square reaches its original position. It now looks the same as the initial position. Thus a square has a rotational symmetry of order 4 about its centre. Observe that the centre of rotation is the centre of the square, the angle of rotation is 90°, the direction is clockwise, and the order is 4. Let us try a quick question. What is the order of rotational symmetry for an equilateral triangle? An equilateral triangle matches itself three times in a full turn, so its order is 3. When rotated by 120°, it looks exactly the same at 3 positions. Now, let us do a hands-on activity. Draw two identical parallelograms, one labelled A B C D on paper and the other A prime B prime C prime D prime on a transparent sheet. Mark the intersection of their diagonals as O and O prime. Place them so A prime lies on A, B prime on B, and so on. O prime falls on O. Stick a pin at point O. Turn the transparent shape clockwise. The shapes coincide twice in a full round, so the order of rotational symmetry is 2.
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Remember, every object has a rotational symmetry of order 1, as it occupies the same position after a rotation of 360°. Such cases have no interest for us. You have around you many shapes which possess rotational symmetry. For example, when you slice certain fruits, the cross-sections are shapes with rotational symmetry. Many road signs also exhibit rotational symmetry. Next time you walk along a busy road, try to identify such signs and find their order of rotational symmetry. Think of more examples and discuss the centre of rotation, the angle of rotation, the direction, and the order. Now, give the order of rotational symmetry for the figures in Figure 12.18 about the marked point. Figure 12.18 (i) has order 2. Figure 12.18 (ii) has order 3. Figure 12.18 (iii) has order 4. Let us move to Exercise 12.2. Question 1 asks which figures have rotational symmetry of order more than 1. Check if each figure looks identical after a rotation of less than 360°. Figures a, c, e, and f will match themselves before a full turn, so they have order more than 1. Figures b and d do not, so they only have order 1. Question 2 asks for the order of rotational symmetry for figures a to h. Figure a matches twice, order 2. Figure b matches twice, order 2. Figure c matches three times, order 3. Figure d matches four times, order 4. Figure e matches four times, order 4. Figure f matches four times, order 4. Figure g matches twice, order 2. Figure h matches three times, order 3.
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Now, let us explore Section 12.4, Line Symmetry and Rotational Symmetry. Some shapes have only line symmetry, some have only rotational symmetry, and some have both. Consider the square in Figure 12.19. It has 4 lines of symmetry and rotational symmetry of order 4. The circle is the most perfect symmetrical figure, because it can be rotated around its centre through any angle and has an unlimited number of lines of symmetry. Every line through the centre, that is every diameter, forms a line of reflectional symmetry, and it has rotational symmetry around the centre for every angle. Some English alphabets have fascinating symmetrical structures. Which capital letters have just one line of symmetry like E? Which have rotational symmetry of order 2 like I? The textbook provides a table for letters Z, S, H, O, E, N, and C. I encourage you to pause and fill in the columns for line symmetry, number of lines, rotational symmetry, and order of rotational symmetry yourself. Now let us solve Exercise 12.3. Question 1 asks for two figures with both line and rotational symmetry. Examples are a square and an equilateral triangle. Question 2 asks for rough sketches. For part i, a triangle with both symmetries of order more than 1 is an equilateral triangle. For part ii, a triangle with only line symmetry is an isosceles triangle. For part iii, a quadrilateral with rotational symmetry of order more than 1 but no line symmetry is a parallelogram. For part iv, a quadrilateral with line symmetry but no rotational symmetry of order more than 1 is an isosceles trapezium or a kite.
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Question 3 asks: If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1? Yes, it should. Question 4 asks you to fill in a table for the centre of rotation, order of rotation, and angle of rotation for a square, rectangle, rhombus, equilateral triangle, regular hexagon, circle, and semi-circle. Use the properties we discussed to find the centre, order, and angle for each shape. Question 5 asks for quadrilaterals with both line and rotational symmetry of order more than 1. These are the square and the rectangle. Question 6 states that after rotating by 60°, a figure looks the same. At what other angles will this happen? It will happen at multiples of 60°, which are 120°, 180°, 240°, 300°, and 360°. Question 7 asks if we can have rotational symmetry of order more than 1 with an angle of rotation of 45°. Yes, because 360 divided by 45 is 8, a whole number. Can it be 17°? No, because 360 divided by 17 is not a whole number. Finally, let us review what we have discussed. A figure has line symmetry if it can be folded along a line so the two parts coincide. Regular polygons have equal sides and equal angles, and multiple lines of symmetry. Each regular polygon has as many lines of symmetry as it has sides. Mirror reflection leads to symmetry, where left-right orientation must be considered. Rotation turns an object about a fixed centre. A half-turn is 180° and a quarter-turn is 90°. If an object looks exactly the same after rotation, it has rotational symmetry. The number of times it matches in 360° is the order. Some shapes have only line symmetry like E, some only rotational like S, and some both like H. The study of symmetry is important for its daily use and beautiful designs.
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Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]