Question number 26. Two regular polygons with a side length of 1 lie on opposite sides of their common side AB. One of them is a regular 15-gon ABCD and the other one is a n-gon ABZY and so on. What value of n makes the distance CZ equal to 1? So drawing a picture to understand the situation we have a red 15-gon here with vertices ABC and then followed by D and so on with side length here equal to 1 and in black a regular n-gon we don't know the value of n with side length always equal to 1 and these two have a side in common side AB. Now following the progression of the vertices here after A and B they split up these are no longer sides in common but between vertex C and Z we have a distance X and we have to figure out the value of n for which that distance X here is equal to 1. So equivalently we can say that X is equal to 1 provided that the triangle BCZ is equilateral or provided that the angle here that I will label as theta everywhere provided that this angle theta is equal to 60 degrees. So we probably should work with angles and so then we recall that the formula here given by 180 degrees minus 360 degrees divided by n is the measure of an interior angle and a regular n-gon. So I mean for example an angle like this here that's that would be an interior angle. So on the right hand side we have 180 degrees in black for the black n-gon minus 360 divided by n and in red on the left hand side we have 180 degrees minus 360 divided by 15 and that we can actually compute that is 180 degrees minus 24 degrees so 156 degrees that's the that's the measure of one of these angles here 156 degrees. And now looking at the vertex B we can see that the central angle around B would be given by 360 degrees that is split up into the interior angle on the 15-gon so 156 degrees plus the angle theta which is 60 degrees plus this unknown quantity 180 minus 360 divided by n and then we can start solving for that so if I subtract theta which is 60 degrees from both sides I get 300 here degrees I get 156 degrees plus 180 degrees minus 360 divided by n and I can keep working with this subtracting 180 subtracting 156 we would obtain that 36 degrees is equal to 360 degrees divided by n and that of course has only one solution when n is equal to 10 so that's that's that we have that n is equal to 10 is the solution that corresponds to the number of sides so the black n-gon is in fact a 10-gon and the answer here is A.

regular polygons

15-gon

n-gon

equilateral triangle

decagon