Find the perimeter of a regular heptagon ABCDEFG if (1/(AE)) + (1/(AC)) = (1/5)


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Question Number 183747 by HeferH last updated on 29/Dec/22

$F i n d t h e p e r i m e t e r o f a r e g u l a r h e p t a g o n$ $A B C D E F G i f \frac{1}{A E} + \frac{1}{A C} = \frac{1}{5}$
	Answered by mr W last updated on 29/Dec/22

	$θ = \frac{5 π}{7}$ $A E = 2 a \sin \frac{θ}{2} = 2 a \sin \frac{5 π}{14}$ $A E = a + 2 a \sin (θ - \frac{π}{2}) = a (1 + 2 \sin \frac{3 π}{14})$ $\frac{1}{2 a \sin \frac{5 π}{14}} + \frac{1}{a (1 + 2 \sin \frac{3 π}{14})} = \frac{1}{5}$ $\frac{a}{5} = \frac{1}{2 \sin \frac{5 π}{14}} + \frac{1}{1 + 2 \sin \frac{3 π}{14}} = 1$ $\Rightarrow a = 5$ $P = 7 a = 35$
	Commented by HeferH last updated on 29/Dec/22

	$t h a n k s :)$
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