10 houses in a circle, different routes, different distances 2003 Portugal p4

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High School Math Grades 9-12, Ages 13-18

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10 houses in a circle, different routes, different distances 2003 Portugal p4

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#1 h Oct 13, 2020, 1:23 PM

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In a village there are only $10$ houses, arranged in a circle of a radius $r$ meters. Each has is the same distance from each of the two closest houses. Every year on Sunday of Pascoa, the village priest makes the Easter visit, leaving the parish house (point $A$ ) and following the path described in Figure 1. This year the priest decided to take the path represented in the Figure 2. Prove that this year the priest will walk another $10r$ meters.

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This post has been edited 2 times. Last edited by parmenides51, Oct 13, 2020, 1:29 PM

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#2 h Oct 13, 2020, 1:24 PM

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#3 h Oct 13, 2020, 2:40 PM

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10AH -10r =(3-4(sin square 36 -0.5(sec36))×10r
=10r

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#4 h Oct 13, 2020, 2:48 PM

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The path in figure 1 is $20r\sin\frac{\pi}{10}$
The path in figure 2 is $20r\sin\frac{3\pi}{10}$
The difference is $40r(\sin 18\cos 36)$
$\cos 36=\frac{\phi}{2}$
$\sin 18=\sqrt{\frac{2-\phi}{4}}$
$\sin 18\cos 36=\frac{1}{4}\sqrt{2\phi^2-\phi^3}=\frac{1}{4}\sqrt{2\phi+2-2\phi-1}=\frac{1}{4}$
So the difference is $10r$

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