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# 10 circle points

Go to solution Solved by k-man,

## Question

there are 10 points on a circle.

a -b = b -c = c -d = d -e = e -f = f -g = g -h = h -i = i -j aprox = j -a

with e being between a and b, f between b, c, g between c, d, h between a, e, i between b, f, and j between c, g.

what's the distance between a and b? assume a circle of radius 10.

Edited by phil1882

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• Solution

16.18

Not sure why the last equality in the OP is approximate.

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Given the radius of the circle as 10, the circumference of the circle is 20π. The distance between A and B is 6π.

Each point is 108° apart from a point labeled with the alphabetic character that is adjacent in the alphabetic sequence, with a and j also 108° degrees apart, and each point being 2π units apart, with a complete circuit being thrice around the circle (three rings ... so send in the clowns.)
Edited by DejMar
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@DejMar,

Looks like you calculated the distance along the circle's curve, while I calculated the distance along the straight line. The OP doesn't explicitly ask for one or the other, but I think unless explicitly stated otherwise, the distance between two points is the shortest distance.

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i *think* though haven't confirmed that the last one, a to j will be slightly shorter than the rest. not by much mind you, but still shorter. never the less you are correct. 16.18, or more accurately, (1+sqrt(5))*5.

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i'll offer a bonus point if you can guess what's significant about these 10 points.

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The points are the vertices of a regular decagon ahebifcjgd. The distances ab, bc, cd, ..., ja are diagonals and are all identical in length. Their length expressed in terms of the radius r of the circumscribed circle is 2*r*sin(54

°).
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a little less obvious, good guess though.

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If the distance between a and j are equal to the distances of of the given pair distances being equal, then the chord length (the straight line between two points on a circle) of AB will equal 10*√(2 - 2*cos(3π/5)), which is approximately 16.18034.

The distance between a and j can also be greater or lesser than the distances between the other points, but without any other information, for those two cases the exact distances between A and B is indeterminable. If the distances are rounded between the other pairs of points after determining the distances between the ten points, then the distance between a and j, of course, will not equal the others.
Edited by DejMar
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Seeing the answer given as (1+sqrt(5))*5, I decided to use the trigonometric identities and a table of exact trigonometric values to express 10·√(2 - 2·cos(3π/5)) without the trigonometric function.

As cos(2u) = 2·cos2(u) - 1, and (3π/5) = 2·(3π/10), u can be equated to (3π/10).

cos(2·((3π/10)) = 2·cos2(3π/10) – 1, which, as cos(3π/10) = √((5 - √5)/8), the expression can be further be simplified as ¼·(1 - √5).

Thus, 10·√(2 - 2·cos(3π/5)) = 5·√(6  + 2·√5) = 5·√(1 + √5)2 = 5·(1  + √5), which is the same as the answer presented in post #5.

Edited by DejMar

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