Page 1 SYMMETRY 265 265 265 265 265 14.1 INTRODUCTION 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. Y ou 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. Y ou might like to recall these ideas. Here are some activities to help you. Chapter 14 Symmetry Nature Architecture Engineering Create some colourful Ink-dot devils Make some symmetrical paper-cut designs. Compose a picture-album showing symmetry. 2020-21 © NCERT not to be republished Page 2 SYMMETRY 265 265 265 265 265 14.1 INTRODUCTION 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. Y ou 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. Y ou might like to recall these ideas. Here are some activities to help you. Chapter 14 Symmetry Nature Architecture Engineering Create some colourful Ink-dot devils Make some symmetrical paper-cut designs. Compose a picture-album showing symmetry. 2020-21 © NCERT not to be republished MATHEMATICS 266 266 266 266 266 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. [Fig 14.1 (i) to (iv)] 14.2 LINES OF SYMMETRY FOR REGULAR POLYGONS Y ou 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° (Fig 14.2). A square is also regular because all its sides are of equal length and each of its angles is a right angle (i.e., 90°). Its diagonals are seen to be perpendicular bisectors of one another (Fig 14.3). Fig 14.1 (i) (ii) (iii) ( i v) 60° 60° 60° a a a Fig 14.2 Fig 14.3 2020-21 © NCERT not to be republished Page 3 SYMMETRY 265 265 265 265 265 14.1 INTRODUCTION 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. Y ou 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. Y ou might like to recall these ideas. Here are some activities to help you. Chapter 14 Symmetry Nature Architecture Engineering Create some colourful Ink-dot devils Make some symmetrical paper-cut designs. Compose a picture-album showing symmetry. 2020-21 © NCERT not to be republished MATHEMATICS 266 266 266 266 266 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. [Fig 14.1 (i) to (iv)] 14.2 LINES OF SYMMETRY FOR REGULAR POLYGONS Y ou 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° (Fig 14.2). A square is also regular because all its sides are of equal length and each of its angles is a right angle (i.e., 90°). Its diagonals are seen to be perpendicular bisectors of one another (Fig 14.3). Fig 14.1 (i) (ii) (iii) ( i v) 60° 60° 60° a a a Fig 14.2 Fig 14.3 2020-21 © NCERT not to be republished SYMMETRY 267 267 267 267 267 If a pentagon is regular, naturally, its sides should have equal length. Y ou will, later on, learn that the measure of each of its angles is 108° (Fig 14.4). A regular hexagon has all its sides equal and each of its angles measures 120°. Y ou will learn more of these figures later (Fig 14.5). 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 [Fig 14.6 (i) - (iv)]. W e 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 (Fig 14.7). A mirror line, thus, helps to visualise a line of symmetry (Fig 14.8). Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes. Fig 14.8 Fig 14.4 Fig 14.5 Fig 14.7 Fig 14.6 2020-21 © NCERT not to be republished Page 4 SYMMETRY 265 265 265 265 265 14.1 INTRODUCTION 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. Y ou 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. Y ou might like to recall these ideas. Here are some activities to help you. Chapter 14 Symmetry Nature Architecture Engineering Create some colourful Ink-dot devils Make some symmetrical paper-cut designs. Compose a picture-album showing symmetry. 2020-21 © NCERT not to be republished MATHEMATICS 266 266 266 266 266 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. [Fig 14.1 (i) to (iv)] 14.2 LINES OF SYMMETRY FOR REGULAR POLYGONS Y ou 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° (Fig 14.2). A square is also regular because all its sides are of equal length and each of its angles is a right angle (i.e., 90°). Its diagonals are seen to be perpendicular bisectors of one another (Fig 14.3). Fig 14.1 (i) (ii) (iii) ( i v) 60° 60° 60° a a a Fig 14.2 Fig 14.3 2020-21 © NCERT not to be republished SYMMETRY 267 267 267 267 267 If a pentagon is regular, naturally, its sides should have equal length. Y ou will, later on, learn that the measure of each of its angles is 108° (Fig 14.4). A regular hexagon has all its sides equal and each of its angles measures 120°. Y ou will learn more of these figures later (Fig 14.5). 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 [Fig 14.6 (i) - (iv)]. W e 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 (Fig 14.7). A mirror line, thus, helps to visualise a line of symmetry (Fig 14.8). Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes. Fig 14.8 Fig 14.4 Fig 14.5 Fig 14.7 Fig 14.6 2020-21 © NCERT not to be republished MATHEMATICS 268 268 268 268 268 While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation, as seen in the figure here (Fig 14.9). The shape is same, but the other way round! Play this punching game! Fold a sheet into two halves Punch a hole two holes about the symmetric fold. Fig 14.10 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 (Fig 14.10). EXERCISE 14.1 1. Copy the figures with punched holes and find the axes of symmetry for the following: (i) (ii) Fig 14.9 R R 2020-21 © NCERT not to be republished Page 5 SYMMETRY 265 265 265 265 265 14.1 INTRODUCTION 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. Y ou 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. Y ou might like to recall these ideas. Here are some activities to help you. Chapter 14 Symmetry Nature Architecture Engineering Create some colourful Ink-dot devils Make some symmetrical paper-cut designs. Compose a picture-album showing symmetry. 2020-21 © NCERT not to be republished MATHEMATICS 266 266 266 266 266 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. [Fig 14.1 (i) to (iv)] 14.2 LINES OF SYMMETRY FOR REGULAR POLYGONS Y ou 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° (Fig 14.2). A square is also regular because all its sides are of equal length and each of its angles is a right angle (i.e., 90°). Its diagonals are seen to be perpendicular bisectors of one another (Fig 14.3). Fig 14.1 (i) (ii) (iii) ( i v) 60° 60° 60° a a a Fig 14.2 Fig 14.3 2020-21 © NCERT not to be republished SYMMETRY 267 267 267 267 267 If a pentagon is regular, naturally, its sides should have equal length. Y ou will, later on, learn that the measure of each of its angles is 108° (Fig 14.4). A regular hexagon has all its sides equal and each of its angles measures 120°. Y ou will learn more of these figures later (Fig 14.5). 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 [Fig 14.6 (i) - (iv)]. W e 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 (Fig 14.7). A mirror line, thus, helps to visualise a line of symmetry (Fig 14.8). Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes. Fig 14.8 Fig 14.4 Fig 14.5 Fig 14.7 Fig 14.6 2020-21 © NCERT not to be republished MATHEMATICS 268 268 268 268 268 While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation, as seen in the figure here (Fig 14.9). The shape is same, but the other way round! Play this punching game! Fold a sheet into two halves Punch a hole two holes about the symmetric fold. Fig 14.10 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 (Fig 14.10). EXERCISE 14.1 1. Copy the figures with punched holes and find the axes of symmetry for the following: (i) (ii) Fig 14.9 R R 2020-21 © NCERT not to be republished SYMMETRY 269 269 269 269 269 2. Given the line(s) of symmetry , find the other hole(s): 3. In the following figures, the mirror line (i.e., the line of symmetry) is given as a dotted line. Complete each figure performing reflection in the dotted (mirror) line. (Y ou might perhaps place a mirror along the dotted line and look into the mirror for the image). Are you able to recall the name of the figure you complete? 4. The following figures have more than one line of symmetry. Such figures are said to have multiple lines of symmetry . Identify multiple lines of symmetry , if any , in each of the following figures: (a) (b) (c) (a) (b) (c) (d) (e) (f) 2020-21 © NCERT not to be republishedRead More

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