Question

A rhombus has an area equal to half the sum of the areas of the squares built on its four sides .What is the ratio of the long diagonal to the short diagonal and how?

Answer 1

let rhombus long diagonal is M , short diagonal is N and side is a

now according to question
area of rhombus =1/2area of square which side is also side of rhombus
=>1/2M x N=1/2 (a^2+a^2+a^2+a^2)=2a^2
=>M x N=4a^2-------(1)
but we know rhombus side and diagonal relation
M^2+N^2=4a^2----------(2)
compare (1) and (2)
M^2+N^2=MN
divided both side by MN and arranged we find
(M/N)^2-(M/N)+1=0
here we see
(M/N) ratio is always complex
i.e M/N=(1+iroot3)/2