A hexagon has all its angles equal, and the lengths of four consecutive sides are 5, 3, 6 and 7, respectively. Find the lengths of the remaining two edges.
Consider the side of size 6 (call it s6) to be horizontal. The unknown side adjacent to the side of s5 is then also horizontal, parallel to the side of s6.
The two sides on either side that connect the s6 to its parallel opposite bring the end of the chain closer to the other side in increments of sqrt(3)/2 compared to the units measuring the sides. So s3 and s5 together mean that the side parallel to s6 is 8*sqrt(3)/2 units away from s6. Since s6 is indeed parallel to its opposite side sqrt(3)/2 times the sum of 7 and the length of the other unknown side must also equal the same value, and in turn the length of the other (not parallel to s6) unknown side must be 1, as 3+5 = 7+1 whether each is multiplied by sqrt(3)/2 or not.
That being the case, if one considers s6 to extend between the origin and (6,0) and s3 begins at the origin while s7 begins at (6,0), the left side of the horizontal unknown begins 1 unit to the right (i.e. x=1) of where s6's leftmost point is. The rightmost point of the horizontal unknown is (71)/2 = 3 units to the right of the rightmost xcoordinate of s6. So the horizontal unknown side has length 6  1 + 3 = 8.

Posted by Charlie
on 20201202 14:33:00 