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.


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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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