Find-the-number-of-sides-of-two-regular-polygons-that-their-sides-has-a-ratio-5-4-and-of-9-as-a-difference-between-their-angles-

Question Number 182474 by Acem last updated on 10/Dec/22

F i n d t h e n u m b e r o f s i d e s o f t w o r e g u l a r p o l y g o n s

t h a t t h e i r s i d e s h a s a r a t i o 5 : 4 a n d o f 9 ° a s a

d i f f e r e n c e b e t w e e n t h e i r a n g l e s .

Answered by mr W last updated on 10/Dec/22

n - s i d e d r e g u l a r p o l y g o n :

a n g l e θ_{n} = 180 - \frac{360}{n}

s a y t w o r e g u l a r p o l y g o n s w i t h x a n d

y s i d e s r e s p e c t i v e l y .

\frac{x}{y} = \frac{5}{4}

θ_{x} - θ_{y} = \frac{360}{y} - \frac{360}{x} = 9

\Rightarrow \frac{1}{y} - \frac{1}{x} = \frac{1}{40}

\Rightarrow \frac{1}{y} - \frac{4}{5 y} = \frac{1}{40} \Rightarrow y = 8 \Rightarrow x = 10

i . e . t h e i r n u m b e r o f s i d e s i s 10 a n d 8

r e s p e c t i v e l y .

Commented by Acem last updated on 10/Dec/22

T h a n k s S i r!

Answered by som(math1967) last updated on 10/Dec/22

l e t s i d e s a r e 5 x, 4 x

\Rightarrow \frac{360}{4 x} - \frac{360}{5 x} = 9

\Rightarrow \frac{90 - 72}{x} = 9

\Rightarrow x = \frac{18}{9} = 2

∴ s i d e s a r e 10, 8

Commented by Acem last updated on 10/Dec/22

T h a n k s s s S i r!

Commented by som(math1967) last updated on 10/Dec/22

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Commented by Acem last updated on 10/Dec/22

C o e u r (:

Answered by Acem last updated on 10/Dec/22

φ_{2} - φ_{1} = \frac{360}{n_{2}} - \frac{360 (\frac{4}{5})}{n_{2}}; φ : e x t e r n a l a n g l e

9 = \frac{72}{n_{2}}, n_{2} = 8, n_{1} = 10

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