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AREAS RELATED TO CIRCLES 227 In a way, we can consider this circular region to be a sector forming an angle of 360° (i.e., of degree measure 360) at the centre O. Now by applying the Unitary Method, we can arrive at the area of the sector OAPB as follows: When degree measure of the angle at the centre is 360, area of the sector = r 2 So, when the degree measure of the angle at the centre is 1, area of the ✁r 2 sector = ✂ 360 Therefore, when the degree measure of the angle at the centre is ✄, area of the 2 ✁r 2 ✝ ☎ ✆ sector = = ✞ ✟ r . 360 360 Thus, we obtain the following relation (or formula) for area of a sector of a circle: Area of the sector of ✄ angle = 360 where r is the radius of the circle and ✄ the angle of the sector in degrees. ✞ 2 r , Now, a natural question arises : Can we find the length of the arc APB corresponding to this sector? Yes. Again, by applying the Unitary Method and taking the whole length of the circle (of angle 360°) as 2 r, we can obtain the required length of the arc APB as 2 360 r ✠ ✡ ☛ . So, length of an arc of a sector of angle ✄ = ✞ 2 360 Now let us take the case of the area of the segment APB of a circle with centre O and radius r (see Fig. 12.7). You can see that : Area of the segment APB = Area of the sector OAPB – Area of ☞ OAB = 2 ✝ ✟ ✞ r – area of ✌ OAB 360 Note : From Fig. 12.6 and Fig. 12.7 respectively, you can observe that : and Area of the major sector OAQB = r 2 – Area of the minor sector OAPB Area of major segment AQB = r 2 – Area of the minor segment APB r . Fig. 12.7
228 MATHEMATICS Let us now take some examples to understand these concepts (or results). Example 2 : Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of the corresponding major sector (Use = 3.14). Solution : Given sector is OAPB (see Fig. 12.8). Area of the sector = = ✁ 2 ✂ ✄r 360 30 3.14 4 4 cm 360 ☎ ☎ ☎ 2 Fig. 12.8 12.56 2 2 = cm ✆ 4.19cm (approx.) 3 Area of the corresponding major sector = r 2 – area of sector OAPB = (3.14 × 16 – 4.19) cm 2 = 46.05 cm 2 = 46.1 cm 2 (approx.) Alternatively, area of the major sector = = = (360 – ) 360 360 30 ✟ 360 ✝ 2 ☎ ✞r 3.14 16 cm ✡ ✠ ☛ ☞ ✌ ☛ 330 ✍ ✎ 2 2 2 3.14 16cm 46.05 cm 360 ☎ ☎ ✆ = 46.1 cm 2 (approx.) Example 3 : Find the area of the segment AYB shown in Fig. 12.9, if radius of the circle is 21 cm and ✏ AOB = 120°. (Use = 22 7 ) Fig. 12.9
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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
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QUADRATIC EQUATIONS 77 Therefore, (
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QUADRATIC EQUATIONS 79 So, 9x 2 - 1
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QUADRATIC EQUATIONS 81 Therefore, t
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QUADRATIC EQUATIONS 83 If b 2 - 4ac
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QUADRATIC EQUATIONS 85 x = 3 25 2 B
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QUADRATIC EQUATIONS 87 Example 15 :
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QUADRATIC EQUATIONS 89 b b b If b 2
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QUADRATIC EQUATIONS 91 EXERCISE 4.4
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ARITHMETIC PROGRESSIONS 5 5.1 Intro
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ARITHMETIC PROGRESSIONS 95 In the e
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ARITHMETIC PROGRESSIONS 97 Similarl
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ARITHMETIC PROGRESSIONS 99 (iv) a 2
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ARITHMETIC PROGRESSIONS 101 Now, lo
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ARITHMETIC PROGRESSIONS 103 Example
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ARITHMETIC PROGRESSIONS 105 It is a
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ARITHMETIC PROGRESSIONS 107 17. Fin
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ARITHMETIC PROGRESSIONS 109 Here, a
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ARITHMETIC PROGRESSIONS 111 Therefo
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ARITHMETIC PROGRESSIONS 113 4. How
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ARITHMETIC PROGRESSIONS 115 EXERCIS
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TRIANGLES 6 6.1 Introduction You ar
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TRIANGLES 119 What can you say abou
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TRIANGLES 121 From the above, you c
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TRIANGLES 123 6.3 Similarity of Tri
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TRIANGLES 125 Therefore, from (1),
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TRIANGLES 127 Example 2 : ABCD is a
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TRIANGLES 129 5. In Fig. 6.20, DE |
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TRIANGLES 131 Let rays BP and CQ in
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TRIANGLES 133 Cut DP = AB and DQ =
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TRIANGLES 135 Now, PQ || EF and ABC
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TRIANGLES 137 Now, BD = 1.2 m × 4
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TRIANGLES 139 Fig. 6.34 2. In Fig.
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TRIANGLES 141 11. In Fig. 6.40, E i
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TRIANGLES 143 Example 9 : In Fig. 6
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TRIANGLES 145 So, from (1) and (2),
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TRIANGLES 147 So, ✁ ✂ ✂ ✂ B
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TRIANGLES 149 or, BL 2 = ✄ AC 2
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TRIANGLES 151 8. In Fig. 6.54, O is
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TRIANGLES 153 (ii) AB 2 = AD 2 - BC
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COORDINATE GEOMETRY 7 7.1 Introduct
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COORDINATE GEOMETRY 157 Therefore,
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COORDINATE GEOMETRY 159 Also, PQ 2
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COORDINATE GEOMETRY 161 So, the req
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COORDINATE GEOMETRY 163 Draw AR, PS
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COORDINATE GEOMETRY 165 Therefore,
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COORDINATE GEOMETRY 167 EXERCISE 7.
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COORDINATE GEOMETRY 169 Example 11
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COORDINATE GEOMETRY 171 EXERCISE 7.
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INTRODUCTION TO TRIGONOMETRY 8 8.1
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INTRODUCTION TO TRIGONOMETRY 175 No
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Page 216 and 217: CIRCLES 211 Theorem 10.2 : The leng
Page 218 and 219: CIRCLES 213 Now, TPR + RPO = 90° =
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Page 246 and 247: SURFACE AREAS AND VOLUMES 241 First
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Page 266 and 267: STATISTICS 261 x = n ✁ fx i 1 n i
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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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