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My question is- ABCDE is e regular pentagon.If AB = 10 then find AC.

Any solution for this question would be greatly appreciated. Thank you,

Hey all thanks for the solutions using trigonometry....can we also get the solution without using trigonometry? –

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hey all thanks for the solutions using trigonometry....can we also get the solution without using trigonometry? –  meg_1997 Jul 13 '12 at 17:32
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2 Answers

Angle ABC = $108^o$, and triangle ABC is isosceles. By the sine rule, $$\frac{\sin(108)}{AC} = \frac{\sin(36)}{10}$$ so $$AC = 10\times\frac{\sin(108)}{\sin(36)} = 5(\sqrt5+1).$$

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This a classical occurrence of the golden ratio $\phi=\frac{1+\sqrt5}2$: the length of the chord $AC$ is $\phi$ times the length of the side $AB$. The following illustration tries to suggest that if you subtract the length of $AB$ from that of $AC$, the remainder has the same ratio to $BC$ as $AB$ had to $AC$, and that you can therefore continue like this subtracting the shorter from the longer length indefinitely. $A$ is the corner on the left, $B$ at the top, $C$ at the right.

pentagon

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