Question

A diagonal of a square has the same length as a diagonal of a rectangle. The area of the rectangle is 60% of the area of the square. What is the ratio of the shorter side to the longer side of the rectangle? Write your answer as a common fraction (or an integer).

How do I even approach this?

Answer 1

let each side of the square be x
then the diagonal is √2 x by Pythagoras

let the short side of the rectangle be a
let the longer side be b
then (1/2)ab = (.6)(1/2)x^2
ab = .6x^2
10ab = 6x^2
x^2 = (5/3)ab

also a^2 + b^2 = 2x^2 , again by Pythagoras, and we are told the diagonals are equal
a^2 + b^2 = 2(5/3)ab
3a^2 + 3b^2 = 10ab
3a^2 - 10ab + 3b^2 = 0
(3a - b)(a - 3b) = 0
a = b/3 or a = 3b, but I defined a < b,
so a = b/3

ratio of a : b = b/3 : b
= 1/3 : 1
= 1 : 3

or a/b = 1/3