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TRIANGLES 127 Example 2 : ABCD is a trapezium with AB || DC. E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB (see Fig. 6.14). Show that AE BF ED FC . Solution : Let us join AC to intersect EF at G (see Fig. 6.15). AB || DC and EF || AB (Given) So, EF || DC (Lines parallel to the same line are parallel to each other) Now, in ✁ ADC, So, EG || DC AE ED = AG GC Similarly, from ✁ CAB, (As EF || DC) (Theorem 6.1) (1) CG AG = CF BF Fig. 6.14 Fig. 6.15 i.e., Therefore, from (1) and (2), AG GC = BF FC AE ED = BF FC (2) Example 3 : In Fig. 6.16, PS SQ = PT TR and ✂ PST = ✂ PRQ. Prove that PQR is an isosceles triangle. Solution : It is given that PS SQ ✄ PT TR So, ST || QR (Theorem 6.2) ☎ Fig. 6.16 Therefore, ✂ PST = ✂ PQR (Corresponding angles) (1)
128 MATHEMATICS So, [From (1) and (2)] Also, it is given that PST = PRQ (2) PRQ = PQR Therefore, PQ = PR (Sides opposite the equal angles) i.e., PQR is an isosceles triangle. EXERCISE 6.2 1. In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii). Fig. 6.17 2. E and F are points on the sides PQ and PR respectively of ✁ a PQR. For each of the following cases, state whether EF || QR : (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm (iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm Fig. 6.18 3. In Fig. 6.18, if LM || CB and LN || CD, prove that AM AB AN AD ✂ ✄ 4. In Fig. 6.19, DE || AC and DF || AE. Prove that BF FE BE EC ☎ Fig. 6.19 ✆
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Foreword Preface Contents 1. Real N
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xi 9. Some Applications of Trigonom
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REAL NUMBERS 1 1.1 Introduction In
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REAL NUMBERS 3 moment we write down
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REAL NUMBERS 5 This algorithm works
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REAL NUMBERS 7 EXERCISE 1.1 1. Use
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REAL NUMBERS 9 An equivalent versio
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REAL NUMBERS 11 So, HCF (6, 72, 120
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3 = a b ✂ REAL NUMBERS 13 Substit
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REAL NUMBERS 15 1.5 Revisiting Rati
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REAL NUMBERS 17 We are now ready to
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REAL NUMBERS 19 You have seen that
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POLYNOMIALS 21 a cubic polynomial a
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POLYNOMIALS 23 Table 2.1 x - 2 -1 0
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POLYNOMIALS 25 Case (iii) : Here, t
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POLYNOMIALS 27 Note that 0 is the o
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a = k, b =- k(✞ + ✟) and c = k
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POLYNOMIALS 31 Example 4 : Find a q
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POLYNOMIALS 33 EXERCISE 2.2 1. Find
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POLYNOMIALS 35 Solution : Note that
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POLYNOMIALS 37 3. If the zeroes of
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PAIR OF LINEAR EQUATIONS IN TWO VAR
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QUADRATIC EQUATIONS 71 Sridharachar
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QUADRATIC EQUATIONS 73 (ii) Since x
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QUADRATIC EQUATIONS 75 Note that we
Page 82 and 83: QUADRATIC EQUATIONS 77 Therefore, (
Page 84 and 85: QUADRATIC EQUATIONS 79 So, 9x 2 - 1
Page 86 and 87: QUADRATIC EQUATIONS 81 Therefore, t
Page 88 and 89: QUADRATIC EQUATIONS 83 If b 2 - 4ac
Page 90 and 91: QUADRATIC EQUATIONS 85 x = 3 25 2 B
Page 92 and 93: QUADRATIC EQUATIONS 87 Example 15 :
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Page 96 and 97: QUADRATIC EQUATIONS 91 EXERCISE 4.4
Page 98 and 99: ARITHMETIC PROGRESSIONS 5 5.1 Intro
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Page 104 and 105: ARITHMETIC PROGRESSIONS 99 (iv) a 2
Page 106 and 107: ARITHMETIC PROGRESSIONS 101 Now, lo
Page 108 and 109: ARITHMETIC PROGRESSIONS 103 Example
Page 110 and 111: ARITHMETIC PROGRESSIONS 105 It is a
Page 112 and 113: ARITHMETIC PROGRESSIONS 107 17. Fin
Page 114 and 115: ARITHMETIC PROGRESSIONS 109 Here, a
Page 116 and 117: ARITHMETIC PROGRESSIONS 111 Therefo
Page 118 and 119: ARITHMETIC PROGRESSIONS 113 4. How
Page 120 and 121: ARITHMETIC PROGRESSIONS 115 EXERCIS
Page 122 and 123: TRIANGLES 6 6.1 Introduction You ar
Page 124 and 125: TRIANGLES 119 What can you say abou
Page 126 and 127: TRIANGLES 121 From the above, you c
Page 128 and 129: TRIANGLES 123 6.3 Similarity of Tri
Page 130 and 131: TRIANGLES 125 Therefore, from (1),
Page 134 and 135: TRIANGLES 129 5. In Fig. 6.20, DE |
Page 136 and 137: TRIANGLES 131 Let rays BP and CQ in
Page 138 and 139: TRIANGLES 133 Cut DP = AB and DQ =
Page 140 and 141: TRIANGLES 135 Now, PQ || EF and ABC
Page 142 and 143: TRIANGLES 137 Now, BD = 1.2 m × 4
Page 144 and 145: TRIANGLES 139 Fig. 6.34 2. In Fig.
Page 146 and 147: TRIANGLES 141 11. In Fig. 6.40, E i
Page 148 and 149: TRIANGLES 143 Example 9 : In Fig. 6
Page 150 and 151: TRIANGLES 145 So, from (1) and (2),
Page 152 and 153: TRIANGLES 147 So, ✁ ✂ ✂ ✂ B
Page 154 and 155: TRIANGLES 149 or, BL 2 = ✄ AC 2
Page 156 and 157: TRIANGLES 151 8. In Fig. 6.54, O is
Page 158 and 159: TRIANGLES 153 (ii) AB 2 = AD 2 - BC
Page 160 and 161: COORDINATE GEOMETRY 7 7.1 Introduct
Page 162 and 163: COORDINATE GEOMETRY 157 Therefore,
Page 164 and 165: COORDINATE GEOMETRY 159 Also, PQ 2
Page 166 and 167: COORDINATE GEOMETRY 161 So, the req
Page 168 and 169: COORDINATE GEOMETRY 163 Draw AR, PS
Page 170 and 171: COORDINATE GEOMETRY 165 Therefore,
Page 172 and 173: COORDINATE GEOMETRY 167 EXERCISE 7.
Page 174 and 175: COORDINATE GEOMETRY 169 Example 11
Page 176 and 177: COORDINATE GEOMETRY 171 EXERCISE 7.
Page 178 and 179: INTRODUCTION TO TRIGONOMETRY 8 8.1
Page 180 and 181: INTRODUCTION TO TRIGONOMETRY 175 No
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INTRODUCTION TO TRIGONOMETRY 177 Fr
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INTRODUCTION TO TRIGONOMETRY 179 Th
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INTRODUCTION TO TRIGONOMETRY 181 EX
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INTRODUCTION TO TRIGONOMETRY 183 As
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INTRODUCTION TO TRIGONOMETRY 185 Ta
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INTRODUCTION TO TRIGONOMETRY 187 1.
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INTRODUCTION TO TRIGONOMETRY 189 Ex
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INTRODUCTION TO TRIGONOMETRY 191 Is
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INTRODUCTION TO TRIGONOMETRY 193 =
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SOME APPLICATIONS OF TRIGONOMETRY 9
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CIRCLES 207 You might have seen a p
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CIRCLES 209 Take a point Q on XY ot
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CIRCLES 211 Theorem 10.2 : The leng
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CIRCLES 213 Now, TPR + RPO = 90° =
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CIRCLES 215 10.4 Summary In this ch
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CONSTRUCTIONS 217 Let us see how th
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CONSTRUCTIONS 219 Solution : Given
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CONSTRUCTIONS 221 Steps of Construc
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AREAS RELATED TO CIRCLES 12 12.1 In
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AREAS RELATED TO CIRCLES 225 You ca
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AREAS RELATED TO CIRCLES 227 In a w
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AREAS RELATED TO CIRCLES 229 Soluti
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AREAS RELATED TO CIRCLES 231 10. An
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AREAS RELATED TO CIRCLES 233 Altern
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AREAS RELATED TO CIRCLES 235 2. Fin
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AREAS RELATED TO CIRCLES 237 11. On
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SURFACE AREAS AND VOLUMES 13 13.1 I
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SURFACE AREAS AND VOLUMES 241 First
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SURFACE AREAS AND VOLUMES 243 Examp
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SURFACE AREAS AND VOLUMES 245 7. A
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SURFACE AREAS AND VOLUMES 247 Examp
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SURFACE AREAS AND VOLUMES 249 How a
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✁ ✂ ✂ ☎ ✄ SURFACE AREAS A
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SURFACE AREAS AND VOLUMES 253 How c
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SURFACE AREAS AND VOLUMES 255 So, t
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SURFACE AREAS AND VOLUMES 257 Now,
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SURFACE AREAS AND VOLUMES 259 3. Gi
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STATISTICS 261 x = n ✁ fx i 1 n i
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STATISTICS 263 Table 14.3 Class int
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STATISTICS 265 Substituting the val
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STATISTICS 267 Example 2 : The tabl
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STATISTICS 269 Example 3 : The dist
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STATISTICS 271 4. Thirty women were
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STATISTICS 273 Clearly, 2 is the nu
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STATISTICS 275 of the students. In
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STATISTICS 277 14.4 Median of Group
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STATISTICS 279 From the table above
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STATISTICS 281 Table 14.14 Marks ob
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STATISTICS 283 Example 7 : A survey
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STATISTICS 285 Solution : Class int
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STATISTICS 287 EXERCISE 14.3 1. The
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STATISTICS 289 5. The following tab
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STATISTICS 291 Remark : Note that b
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STATISTICS 293 EXERCISE 14.4 1. The
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PROBABILITY 15 The theory of probab
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PROBABILITY 297 For another example
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PROBABILITY 299 Solution : Kritika
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PROBABILITY 301 That is, the probab
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PROBABILITY 303 Example 7 : There a
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PROBABILITY 305 Let E be the event
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PROBABILITY 307 4 6 5 4 6 5 1 2 3 4
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PROBABILITY 309 10. A piggy bank co
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PROBABILITY 311 25. Which of the fo
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ANSWERS/HINTS EXERCISE 1.1 1. (i) 4
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ANSWERS/HINTS 347 his daughter. To
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ANSWERS/HINTS 349 EXERCISE 3.5 1. (
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ANSWERS/HINTS 351 5. Marks in mathe
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ANSWERS/HINTS 353 n 4. 12. By putti
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ANSWERS/HINTS 355 3. 61m; 5th line
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ANSWERS/HINTS 357 5. 40 3 m 6. 19 3
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ANSWERS/HINTS 359 EXERCISE 13.3 1.
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ANSWERS/HINTS 361 EXERCISE 15.1 1.
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ANSWERS/HINTS 363 ✆ ✂ ✂ ✂ 2
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PROOFS IN MATHEMATICS A1 A1.1 Intro
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PROOFS IN MATHEMATICS 315 (v) Some
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PROOFS IN MATHEMATICS 317 used to d
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PROOFS IN MATHEMATICS 319 Number of
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PROOFS IN MATHEMATICS 321 4. Using
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PROOFS IN MATHEMATICS 323 Remark :
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PROOFS IN MATHEMATICS 325 For examp
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PROOFS IN MATHEMATICS 327 In genera
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PROOFS IN MATHEMATICS 329 2. Write
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PROOFS IN MATHEMATICS 331 Since np
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PROOFS IN MATHEMATICS 333 EXERCISE
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MATHEMATICAL MODELLING 335 (v) Pred
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MATHEMATICAL MODELLING 337 For inst
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MATHEMATICAL MODELLING 339 A2.3 Som
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MATHEMATICAL MODELLING 341 Example
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MATHEMATICAL MODELLING 343 3. A T.V
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