SYMMETRY AND PRACTICAL GEOMETRY

Transcription

1 MTHEMTIS UNIT 9 SYMMETRY N RTIL GEOMETRY () Main oncepts and Resuts figure is said to have ine symmetry, if by foding the figure aong a ine, the eft and right parts of it coincide exacty. The ine is caed the ine (or axis) of symmetry of the figure. figure may have no ine of symmetry, one ine of symmetry, two ines of symmetry, three ines of symmetry and so on. Line symmetry is cosey reated to mirror refection. The distance of the image of a point (or object) from the ine of symmetry (mirror) is the same as that of the point from that ine of symmetry. Many constructions can be made using different instruments of a geometry box. () Soved Exampes In exampes 1 and 2, out of four given options, ony one is correct. Write the correct answer. Exampe 1: Which of the foowing etters does not have any ine of symmetry? () E () T () N () X orrect answer is ()

2 UNIT-9 Exampe 2: Which of the foowing anges cannot be constructed using ruer and compasses? () 75 () 15 () 135 () 85 orrect answer is () In exampes 3 to 5, fi in the banks so that the statements are true: Exampe 3: If is the image of in ine and is the image of in ine, then =. Exampe 4: In Fig. 9.1, the ine segments Q and RQ have been marked on a ine such that Q = and RQ =. Then =. R Fig. 9.1 Q R Exampe 5: The number of scaes in a protractor for measuring the anges is. Two In exampes 6 and 7, state whether the statements are true or fase: Exampe 6: Using the set squares and , we can draw an ange of 75. True. (Since 75 = ) Exampe 7: Exampe 8. circe has ony 8 ines of symmetry. Fase ( circe has infinitey many ines of symmetry). Write the etters of the word LGER which have no ine of symmetry. The etters L, G and R have no ine of symmetry. (o you see why the dotted ine is not the ine of symmetry in?) 134 EXEMLR ROLEMS

3 MTHEMTIS Exampe 9: raw a ine segment equa to the sum of two ine segments given in Fig raw a ine and on it, cut a ine segment Q =, using compasses. ( Fig. 9.3 ) Q 2. With Q as centre and Fig. 9.3 as radius, draw an arc to cut a ine segment QS = on as shown in Q Fig Then, ine segment Fig. 9.4 S is equa to the sum of and, i.e., S = + Fig. 9.2 S Exampe 10. raw an ange equa to the difference of two anges given in Fig E F R Q Fig raw an ange equa to EF (as EF > QR), using ruer and compasses. 2. With as one of the arms, draw an ange S equa to QR such that S is in the interior of as shown in Fig Then, S is the required S ange which is equa to EF QR. [Note: For making S = EF QR, how wi you draw ray S?] Fig. 9.6 SYMMETRY N RTIL GEOMETRY 135

4 UNIT-9 Exampe 11. ompete Fig. 9.7 so that is the ine of symmetry of the competed figure. Fig. 9.7 The figure can be competed as shown in Fig. 9.8, by drawing the points symmetric to different corners(points) with respect to ine. Fig. 9.8 () Exercise In questions 1 to 17, out of the given four options, ony one is correct. Write the correct answer. 1. In the foowing figures, the figure that is not symmetric with respect to any ine is: (i) (ii) (iii) (iv) () (i) () (ii) () (iii) () (iv) 2. The number of ines of symmetry in a scaene triange is () 0 () 1 () 2 () EXEMLR ROLEMS

5 MTHEMTIS 3. The number of ines of symmetry in a circe is () 0 () 2 () 4 () more than 4 4. Which of the foowing etters does not have the vertica ine of symmetry? () M () H () E () V 5. Which of the foowing etters have both horizonta and vertica ines of symmetry? () X () E () M () K 6. Which of the foowing etters does not have any ine of symmetry? () M () S () K () H 7. Which of the foowing etters has ony one ine of symmetry? () H () X () Z () T 8. The instrument to measure an ange is a () Ruer () rotractor () ivider () ompasses 9. The instrument to draw a circe is () Ruer () rotractor () ivider () ompasses 10. Number of set squares in the geometry box is () 0 () 1 () 2 () The number of ines of symmetry in a ruer is () 0 () 1 () 2 () The number of ines of symmetry in a divider is () 0 () 1 () 2 () The number of ines of symmetry in compasses is () 0 () 1 () 2 () The number of ines of symmetry in a protractor is () 0 () 1 () 2 () more than 2 SYMMETRY N RTIL GEOMETRY 137

6 UNIT The number of ines of symmetry in a 45 o - 45 o - 90 o set-square is () 0 () 1 () 2 () The number of ines of symmetry in a 30 o - 60 o - 90 o set square is () 0 () 1 () 2 () The instrument in the geometry box having the shape of a triange is caed a () rotractor () ompasses () ivider () Set-square In questions 18 to 42, fi in the banks to make the statements true. 18. The distance of the image of a point (or an object) from the ine of symmetry (mirror) is as that of the point (object) from the ine (mirror). 19. The number of ines of symmetry in a picture of Taj Maha is. 20. The number of ines of symmetry in a rectange and a rhombus are (equa/unequa). 21. The number of ines of symmetry in a rectange and a square are (equa/unequa). 22. If a ine segment of ength 5cm is refected in a ine of symmetry (mirror), then its refection (image) is a of ength. 23. If an ange of measure 80 o is refected in a ine of symmetry, then the refection is an of measure. 24. The image of a point ying on a ine with respect to the ine of symmetry ies on. 25. In Fig. 9.10, if is the image of the point with respect to the ine and is any point ying on, then the engths of ine segments and are. Fig EXEMLR ROLEMS

7 MTHEMTIS 26. The number of ines of symmetry in Fig is. Fig The common properties in the two set-squares of a geometry box are that they have a ange and they are of the shape of a. 28. The digits having ony two ines of symmetry are and. 29. The digit having ony one ine of symmetry is. 30. The number of digits having no ine of symmetry is. 31. The number of capita etters of the Engish aphabets having ony vertica ine of symmetry is. 32. The number of capita etters of the Engish aphabets having ony horizonta ine of symmetry is. 33. The number of capita etters of the Engish aphabets having both horizonta and vertica ines of symmetry is. 34. The number of capita etters of the Engish aphabets having no ine of symmetry is. 35. The ine of symmetry of a ine segment is the bisector of the ine segment. 36. The number of ines of symmetry in a reguar hexagon is. 37. The number of ines of symmetry in a reguar poygon of n sides is. 38. protractor has ine/ines of symmetry. SYMMETRY N RTIL GEOMETRY 139

8 UNIT o - 60 o - 90 o set-square has ine/ines of symmetry o - 45 o - 90 o set-square has ine/ines of symmetry. 41. rhombus is symmetrica about. 42. rectange is symmetrica about the ines joining the of the opposite sides. In questions 43-61, state whether the statements are true (T) or fase (F). 43. right triange can have at most one ine of symmetry. 44. kite has two ines of symmetry. 45. paraeogram has no ine of symmetry. 46. If an isoscees triange has more than one ine of symmetry, then it need not be an equiatera triange. 47. If a rectange has more than two ines of symmetry, then it must be a square. 48. With ruer and compasses, we can bisect any given ine segment. 49. Ony one perpendicuar bisector can be drawn to a given ine segment. 50. Two perpendicuars can be drawn to a given ine from a point not ying on it. 51. With a given centre and a given radius, ony one circe can be drawn. 52. Using ony the two set-squares of the geometry box, an ange of 40 o can be drawn. 53. Using ony the two set-squares of the geometry box, an ange of 15 o can be drawn. 54. If an isoscees triange has more than one ine of symmetry, then it must be an equiatera triange. 55. square and a rectange have the same number of ines of symmetry. 56. circe has ony 16 ines of symmetry o - 45 o - 90 o set-square and a protractor have the same number of ines of symmetry. 140 EXEMLR ROLEMS

9 MTHEMTIS 58. It is possibe to draw two bisectors of a given ange. 59. reguar octagon has 10 ines of symmetry. 60. Infinitey many perpendicuars can be drawn to a given ray. 61. Infinitey many perpendicuar bisectors can be drawn to a given ray. 62. Is there any ine of symmetry in the Fig. 9.12? If yes, draw a the ines of symmetry. Fig In Fig. 9.13, QRS is a rectange. State the ines of symmetry of the rectange. Q S R Fig SYMMETRY N RTIL GEOMETRY 141

10 UNIT Write a the capita etters of the Engish aphabets which have more than one ines of symmetry. 65. Write the etters of the word MTHEMTIS which have no ine of symmetry. 66. Write the number of ines of symmetry in each etter of the word SYMMETRY. 67. Match the foowing: Shape Number of ines of symmetry (i) Isoscees triange (a) 6 (ii) Square (b) 5 (iii) Kite (c) 4 (iv) Equiatera triange (d) 3 (v) Rectange (e) 2 (vi) Reguar hexagon (f) 1 (vii) Scaene triange (g) Open your geometry box. There are some drawing toos. Observe them and compete the foowing tabe: Name of the too Number of ines of symmetry (i) The Ruer (ii) The ivider (iii) The ompasses (iv) The rotactor (v) Trianguar piece with two equa sides (vi) Trianguar piece with unequa sides 142 EXEMLR ROLEMS

11 MTHEMTIS 69. raw the images of points and in ine of Fig and name them as and respectivey. Measure and. re they equa? Fig In Fig. 9.15, the point is the image of point in ine and ine segment intersects the ine at. (a) Is the image of in ine the point itsef? (b) Is =? (c) Is + = +? (d) Is that point on ine from which the sum of the distances of points and is minimum? Fig ompete the figure so that ine becomes the ine of symmetry of the whoe figure (Fig. 9.16). Fig raw the images of the points, and in the ine m (Fig. 9.17). Name them as, and, respectivey and join m them in pairs. Measure,,,, and. Is =, = and =? Fig SYMMETRY N RTIL GEOMETRY 143

12 UNIT raw the images, Q and R of the points, Q and R, respectivey in the ine n (Fig. 9.18). n Join Q and Q R to form an ange Q R. Measure QR and Q R. re the two anges equa? Q R 74. ompete Fig by taking as the ine of symmetry of the whoe figure. Fig Fig raw a ine segment of ength 7cm. raw its perpendicuar bisector, using ruer and compasses. 76. raw a ine segment of ength 6.5cm and divide it into four equa parts, using ruer and compasses. 77. raw an ange of 140 o with the hep of a protractor and bisect it using ruer and compasses. 78. raw an ange of 65 o and draw an ange equa to this ange, using ruer and compasses. 79. raw an ange of 80 o using a protractor and divide it into four equa parts, using ruer and compasses.heck your construction by measurement. 80. opy Fig on your notebook and draw a perpendicuar to through, using (i) set squares (ii) rotractor (iii) ruer and compasses. How many such perpendicuars are you abe to draw? Fig EXEMLR ROLEMS

13 MTHEMTIS 81. opy Fig on your notebook and draw a perpendicuar from to ine m, using (i) set squares (ii) rotractor (iii) ruer and compasses. How many such perpendicuars are you abe to draw? Fig m 82. raw a circe of radius 6cm using ruer and compasses. raw one of its diameters. raw the perpendicuar bisector of this diameter. oes this perpendicuar bisector contain another diameter of the circe? 83. isect XYZ of Fig Z X Y Fig raw an ange of 60 o using ruer and compasses and divide it into four equa parts. Measure each part. 85. isect a straight ange, using ruer and compasses. Measure each part. 86. isect a right ange, using ruer and compasses. Measure each part. isect each of these parts. What wi be the measure of each of these parts? 87. raw an ange of measure 45 o, using ruer and compasses. Now draw an ange of measure 30 o, using ruer and compasses as shown in Fig What is the measure of? 30 O 45 O Fig raw a ine segment of ength 6cm. onstruct its perpendicuar bisector. Measure the two parts of the ine segment. 89. raw a ine segment of ength 10cm. ivide it into four equa parts. Measure each of these parts. SYMMETRY N RTIL GEOMETRY 145

14 UNIT-9 () ctivities ctivity 1: ctivity 2: Make three different ink bot devis in your notebook and mark their ine of symmetry. raw a the ines of symmetry of Fig by paper foding. Fig ctivity 3: raw an ange of 15 o by first drawing an ange of 60 o and then an ange of 45 o, using ruer and compasses. ctivity 4: Using ruer and compasses draw an ange of 90 o and in its interior, draw two rays with the initia point of each as the vertex of the ange so that each of the three anges so formed is of 30 o (See Fig. 9.25). 30 O 30 O 30 O Fig ctivity 5: ctivity 6: ctivity 7: ctivity 8: raw an ange of 45 o and in its interior, draw two rays to form three anges each of measure 15 o, using ruer and compasses. raw an ange of 135 o and in its interior, draw two rays to form three anges each of equa measure, using ruer and compasses. raw the perpendicuar bisectors of, and (Fig. 9.26). What do you observe? Fig isect E and E by drawing up their perpendicuar bisectors in (Fig. 9.27). Let be the point of intersection of these perpendicuar bisectors check E whether = E, E = Fig EXEMLR ROLEMS

15 ctivity 9: isect and by drawing their perpendicuar bisectors (Fig. 9.28). Make the point of intersecton as. heck whether = = MTHEMTIS ctivity 10: raw two ine segments of engths 8cm and 6cm. Using these ine segments, construct a ine segment of ength (8 + 6)cm. ctivity 11: raw two ine segments of engths 3cm and 5cm. onstruct ine segments of the foowing engths using these ine segments: (a) 6cm (b) 15cm (c) (3+5)cm (d) (6+5)cm (e) (9 5)cm (f) (5 3)cm ctivity 12: raw two ine segments of engths 3cm and 6cm. onstruct ine segments, equa to the foowing engths, using these ine segments. (a) cm (b) 2 ctivity 13: rop perpendicuars from to and from to (Fig. 9.29) cm (c) ( ) 2 Fig Fig cm ctivity 14: O is the centre of the circe (Fig. 9.30). rop perpendicuar from on. Where does it meet? O ctivity 15: opy the figure and bisect and (Fig. 9.31). Let the bisectors meet at some point. Measure ange. Fig Fig SYMMETRY N RTIL GEOMETRY 147

16 UNIT-9 ctivity 16: Fig (a) isect ange 1 and ange 2 (Fig. 9.32). (b) Measure the ange between these bisectors. (c) Now bisect ange 3 and ange 4. (d) Measure the ange formed between these bisectors. (e) What do you obeserve from (b) and (d)? an you concude something? ctivity 17: onstruct an ange equa to Fig. 9.33, using ruer and compasses times the QR of R Q Fig ctivity 18: isect ange, ange and ange (Fig. 9.34). What do you observe? Fig EXEMLR ROLEMS