Convex hexagon ABCDEF is equiangular but has no two sides the same length. Its sides in some order are 1, 2, 3, 4, 5 and 6 units long. If AB=1 and CD>BC, what are the lengths of BC, CD, DE, EF and FA?

Another convex hexagon is also equiangular and has sided measuring 1, X, 3, 4, 5, and 6 units long in that order going clockwise. What is the measure of X?

Since it is equiangular, that means the angle between two adjacent sides will always be 120 degrees.

To solve the second problem, I think I will simply map out the coordinates of the vertices. Then using the two vertices on either end of X, I can calculate the length of X (BC). For my own brain organization, I will start with the 3 unit edge length, which is side CD.

I will place CD on the x-axis, with C on the origin. So the coordinates are C = (0,0) and D = (3, 0).

Moving clockwise, E will be "down and to the right" of D. So E in relation to D is simply 4 units away at a 120 degree angle from edge CD (or a -60 degree angle from the x-axis). So the coordinates of E are (3 + 4cos(-60), 0 + 4sin(-60)) = (3+4/2, 4(-sqrt(3)/2)) = (5, -2sqrt(3))

Next, F will be "down and to the left" of E. So F in relation to E is simply 5 units away at a 120 degree angle from edge DE (or a –120 degree angle from the x-axis). So the coordinates of F are (5+5cos(-120), -2sqrt(3)+5sin(-120)) = (5-5/2, -2sqrt(3)+5(-sqrt(3)/2)) = (5/2, -9/2*sqrt(3))

A will be directly to the left of F. So A in relation to F is simply 6 units away at a 120 degree angle from edge EF (or a 180 degree angle from the x-axis). So the coordinates of A are (5/2+6*cos(180), -9/2*sqrt(3)+6sin(180)) = (5/2+6(-1), -9/2*sqrt(3)+0) = (-7/2, -9/2*sqrt(3))

B will be "up and to the left" of A. So B in relation to A is simply 1 unit away at a 120 degree angle from edge FA (or a 120 degree angle from the x-axis). So the coordinates of B are (-7/2+1cos(120), -9/2*sqrt(3)+1sin(120)) = (-7/2-1/2, -9/2*sqrt(3)+sqrt(3)/2)) = (-4, -4sqrt(3))

So the distance from B to C is X, where X^2 = (0-(-4))^2 + (0-(-4sqrt(3)))^2 = 4^2 + 4^2*3 = 64.

So X = 8

Posted by nikki on 2004-10-15 14:34:16