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The interior angles of a polygon are in AP. The smallest angle is 120° and common sense difference is is 5° find the no. ofsides of polygon

The interior angles of a polygon are in AP.  The smallest angle is 120° and common sense difference is is 5° find the no.  ofsides of polygon
 

Grade:12th pass

2 Answers

Arun
25763 Points
3 years ago
The sum of the angles in an n-sided shape is (n-2)*180. We know: 

120 + 125 + 130 + ... = (n-2)*180 

If we have n angles, the largest angle will be 120 + (n-1)*5 = 115 + 5n. We're adding an evenly spaced series here, so we can find the sum, using the average fomula: 

sum = (average)*(number) 

Because the series is evenly spaced, the average of the series is equal to the average of the smallest (120) and the largest (115 + 5n). There are n terms, so:

sum = [(120 + 115 + 5n)/2]*n 
(n-2)*180 = (235 + 5n)*n/2 
360n - 720 = 235n + 5n^2 
72n - 144 = 47n + n^2 
0 = n^2 - 25n + 144 
0 = (n-16)(n-9) 
n = 9 or n = 16. 

We must discard the n=16 solution, because if n=16, the angles go past 180, and you can't have a 180 degree angle as an interior angle in a polygon- it makes a straight line.
sagar kumar
99 Points
3 years ago

The smallest interior angle is 120 deg. and the corresponding exterior angle is 60 deg. The sum of the exterior angles of the polygon is 360, and the exterior angles are in an AP.

Sn = (n/2)[2a +(n-1)d]

360 = (n/2)[(2x60) +(n-1)x(-5)], or

720 = n[120 -5n + 5], or

720 = n[-5n + 125], or

144 = n[-n +25], or

n^2 - 25n + 144 = 0, or

(n-9)(n-16) = 0

or n = 9. The figure is an irregular nonagon. The sum of the nine exterior angles which are 60,55,50,45,40,35,30,25,20 = 360.

Check: Sn = (9/2)[2*20 + (9–1)*5] = (9/2)[40 +8*5]] = (9/2)[40+40] = 9*80/2 = 360. Correct.

n = 9. The figure is an irregular nonagon.

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