Ex 9.2, 18 - The difference between two interior angles - Arithmetic Progression (AP): Statement

  1. Chapter 9 Class 11 Sequences and Series
  2. Serial order wise
Ask Download

Transcript

Ex 9.2 , 18 The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon. Difference between consecutive angles = 5° Smallest angle = 120° Second smallest angle = 120° + 5° = 125° Third smallest angle = 125° + 5° = 130° Thus, the angles are 120, 125,130,…. The angles form an A.P. as difference of consecutive terms is constant. Here, first term = a = 120 Common difference = 125 – 120 For A.P., Sn = n/2 ( 2a + (n – 1)d ) Where, Sn = sum of n terms of A.P. n = number of terms a = first term and d = common difference Here, Sn = Sum of all angles of polygon , a = 120, d = 5 Sn = n/2 (2(120) + (n – 1)5) Also, Sum of all angles of a polygon with n sides= 180°(n – 2). ∴ Sn = 180 (n – 2) Comparing (1) & (2) n/2 (2(120) + (n – 1)5) = 180 (n – 2) n[240 + (n – 1)5] = 180 × 2(n – 2) n[240 + (n – 1)5 ] = 360(n – 2) 240n + 5n2 – 5n = 360n – 720 5n2 + 235n – 360n + 720 = 0 5n2 – 125n + 720 = 0 5(n2 – 25n + 144) = 0 (n2 – 25n + 144) = 0/5 n2 – 25n + 144 = 0 n2 – 16n – 9n + 144 = 0 n(n – 16) – 9 (n – 16) = 0 (n – 9) (n – 16) = 0 n = 9 or 16 ∴ Number of sides = 9 or 16

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail