Regular hexagon problem in similarity chapter

[Teachmmburns]

Please help me solve this problem. I know I need to write systems of equations to solve for x and y (x=EJ=KB and y=JO=OK). I have one equation x+y=12, but cannot figure out another equation to write to complete my system of equations to solve for x and y.

See attached image for problem description:

 

[HallsofIvy]

It really doesn't make sense to say "I have one equation x+y=12" without saying what coordinate system you are using. I would set up a coordinate system so that the line segment EB lay along the x-axis and the center of the hexagon, O, is the origin. Is that what you did? Now, the coordinates of the points are A= (6, 6 sqrt(3)), B= (12, 0), C= (6, -6 sqrt(3)), D= (-6, -6 sqrt(3)), E= (-12, 0), F= (-6, 6 sqrt(3)). G is the midpoint of BC so its coordinates are ((12+ 6)/2, -6 sqrt(3)/2)= (9, -3 sqrt(3)). H is the midpoint of DE so its coordinates are (-9, -3 sqrt(3)).

From that you can find the equations of line FG, the line through (-6, 6 sqrt(3)) and (9, -3sqrt(3)), and HA, the line through (-9, -3 sqrt(3)) and (6, 6 sqrt(3)) then find where those lines intersect the x- axis (where y= 0).

 

[Teachmmburns]

It really doesn't make sense to say "I have one equation x+y=12" without saying what coordinate system you are using. I would set up a coordinate system so that the line segment EB lay along the x-axis and the center of the hexagon, O, is the origin. Is that what you did? Now, the coordinates of the points are A= (6, 6 sqrt(3)), B= (12, 0), C= (6, -6 sqrt(3)), D= (-6, -6 sqrt(3)), E= (-12, 0), F= (-6, 6 sqrt(3)). G is the midpoint of BC so its coordinates are ((12+ 6)/2, -6 sqrt(3)/2)= (9, -3 sqrt(3)). H is the midpoint of DE so its coordinates are (-9, -3 sqrt(3)).

From that you can find the equations of line FG, the line through (-6, 6 sqrt(3)) and (9, -3sqrt(3)), and HA, the line through (-9, -3 sqrt(3)) and (6, 6 sqrt(3)) then find where those lines intersect the x- axis (where y= 0).

You don't need to use a coordinate system - this is for high school geometry in a similarity section where you use proportions to solve for unknown lengths. Therefore, it is not necessary to put shapes on a coordinate system to solve this problem.

I defined x to be equal to the line segment EJ and KB as they are congruent in the image given, and y to be equal to the line segment JO and OK as they are also congruent.

EB is 24 in total length which is how I got x+y=12 ; reduced from 2x+2y=24.