phil1882 13 Posted December 9, 2014 Report Share Posted December 9, 2014 (edited) 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 December 9, 2014 by phil1882 Quote Link to post Share on other sites

0 Solution k-man 26 Posted December 9, 2014 Solution Report Share Posted December 9, 2014 16.18 Not sure why the last equality in the OP is approximate. Quote Link to post Share on other sites

0 DejMar 9 Posted December 10, 2014 Report Share Posted December 10, 2014 (edited) 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 December 10, 2014 by DejMar Quote Link to post Share on other sites

0 k-man 26 Posted December 11, 2014 Report Share Posted December 11, 2014 @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. Quote Link to post Share on other sites

0 phil1882 13 Posted December 12, 2014 Author Report Share Posted December 12, 2014 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. Quote Link to post Share on other sites

0 phil1882 13 Posted December 12, 2014 Author Report Share Posted December 12, 2014 i'll offer a bonus point if you can guess what's significant about these 10 points. Quote Link to post Share on other sites

0 k-man 26 Posted December 12, 2014 Report Share Posted December 12, 2014 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°). Quote Link to post Share on other sites

0 phil1882 13 Posted December 12, 2014 Author Report Share Posted December 12, 2014 a little less obvious, good guess though. Quote Link to post Share on other sites

0 DejMar 9 Posted December 16, 2014 Report Share Posted December 16, 2014 (edited) 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 December 16, 2014 by DejMar Quote Link to post Share on other sites

0 DejMar 9 Posted December 16, 2014 Report Share Posted December 16, 2014 (edited) 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·cos^{2}(u) - 1, and (3π/5) = 2·(3π/10), u can be equated to (3π/10). cos(2·((3π/10)) = 2·cos^{2}(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 December 16, 2014 by DejMar Quote Link to post Share on other sites

## Question

## phil1882 13

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## Link to post

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