Question

Please do all these 4 questions if you don't know all the 4 questions then try to solve at least 1 2.....
Please tomorrow is my exam try to solve my questions......

Answer 1

19) ratio of sides = 1:2
     ratio of interior angles = 3:8
Let the sides be n1 , n2 for the 2 polygons.
Interior angle sum = (n-2)*180
n1/n2 = 1/2... n2 = 2n1...(i)

(n1-2)*180/(n2-2)*180 = 3/8
n1-2/n2-2 = 3/8
8n1 -16 = 3n2 - 6
8n1 - 3n2 = 10.....(ii)
(i) in (ii)
8n1 - 6n1 = 10
n1 = 5
n2 =10..
Hence sides of polygon are 5,10 respectively.
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20) Given: n1/n2 = 4/5
n1 = 4*n2/5.....(i)
Also, interior angle = (n-2)*180/n

(n1-2)*180/n1 : (n2-2)*180/n2 = 15:16
(n1-2)*n2/(n2-1)*n1 = 15/16
16n1n2 - 32n2 = 15n1n2 - 15n1....(ii)

Sub for n1 from (i)

64n2*n2/5 - 32n2 = 60n2*n2/5 - 60n2/5
64n2*n2 - 160n2 = 60n2*n2 - 60n2  [took 5 as the lcm and '5' from denominator got cancelled after taking lcm]

4n2*n2 = 100n2
4n2 = 100
n2 = 25
n1 = 4n2/5 = 20
Hence sides are 20,25 respectively.
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21) Given : 360/n-1 - 360/n+1 = 9
360*[1/n-1 - 1/n+1] =9
[n+1 - (n-1)]/(n+1)*(n-1) = 1/40
2/n^2-1 = 1/40
n^2 - 1 = 80
n^2 = 81
n=9
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22) Given:
360/(n-2) - 360/(n+1) = 27
360*[1/n-2 - 1/n+1] = 27
[n+1-n+2]/(n^2-n-2) = 3/40
3/n^2-n-2 = 3/40
n62-n-2 = 40
n^2-n-42 = 0
n^2 - 7n + 6n -42 = 0
(n-7)*(n+6) =0
Hence, n=7
Sides are 5 & 8 respectively
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Formulas: Sum of interior angles of regular polygon(n sides) = (n-2)*180
Each interior angle = (n-2)*180/n
Each exterior angle = 360/n

Hope it helps