SOLUTION: A regular hexagon and a rectangle have the same perimeter, P. A side of the hexagon is 4 less than the length, l, of the rectangle. The width of the rectangle, w, is 2 less than th

Question 835030: A regular hexagon and a rectangle have the same perimeter, P. A side of the hexagon is 4 less than the length, l, of the rectangle. The width of the rectangle, w, is 2 less than the length of the rectangle. What is the perimeter of the hexagon?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let H represent the perimeter of the hexagon.
let R represent the perimeter of the rectangle.
you get 2 equations.
H = 6S
S represents the length of a side of the hexagon.
R = 2L + 2W
L represents the length of the rectangle and W represents the width of the rectangle.
those are the formulas for the perimeter of the hexagon and the perimeter of the rectangle respectively.
since the perimeters are the same, we have:
H = R
replacing H and R with their equivalent values of H = 6S and R = 2L + 2W, we have:
6S = 2L + 2W
we are given the following:
S = L - 4
W = L - 2
in the equation of 6S = 2L + 2W, we can replace S with L - 4 and W with L - 2 and then we can solve for L.
6S = 2L + 2W becomes:
6(L-4) = 2L + 2(L-2) after we do the replacement.
simplify this equation to get:
6L - 24 = 2L + 2L - 4
combine like terms to get:
6L - 24 = 4L - 4
subtract 4L from both sides of this equation and add 24 to both sides of this equation to get:
2L = 20
divide both sides of this equation by 2 to get:
L = 10
this makes S = 10 - 4 = 6
this also makes W = 10 - 2 = 8
we have:
L = 10
W = 8
S = 6
to confirm we have the right solution, we go back to the original equation to see if it is true once we replace the variables with their values.
the original equation is:
H = R which we translated to:"
6S = 2L + 2W
replacing the variables with their values gets us:
6*6 = 2*10 + 2*8
simplify this to get:
36 = 20 + 16
simplify further to get:
36 = 36
this confirms the solution is good.
the perimeter of the hexagon is equal to the perimeter of the rectangle which is equal to 36.

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