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Answer 1 · 2018-06-22T02:25:38

#:. "Ratio of longer to shorter side of rectangle " = 3 + 2 sqrt2#

#"Perimeter of rectangle "= 2l + 2w#

#"Perimeter of rhombus "= 4a#

#2l + 2w = 4a " or " l + w = 2a#

#"Area of rectangle " = lw#

#"Area of rhombus "= a^2 sin theta#

#l * w = a^2 sin 30= a^2 / 2#

#(l+w)^2 = 4a^2#

#(l + w)^2 / (lw) = (4a^2) / (a^2 / 2)#

#(l+w)^2 = 8lw#

#l^2 / (lw) - 6 + w^2 / (lw) = 0#

#(l/w) - 6 + 1/(l/w) = 0#

#l/w = (6 +- sqrt(6^2 - 4)) / 2 = (6 +- sqrt32) / 2 = 3 +- 2sqrt2#

#:. "Ratio of longer to shorter side of the rectangle is "#

#= 3 + 2 sqrt2, 3 - 2sqrt2#