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?

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

To approach this problem, we can start by assigning some variables. Let's say the side length of the square is "s" and the longer side of the rectangle is "l."

Since the diagonal of a square is equal to the length of its side multiplied by √2, we can write the equation as:

s√2 = l√2

Simplifying, we get:

s = l

We also know that the area of the rectangle is 60% of the area of the square. The area of the square is s^2, so the area of the rectangle is 0.6s^2.

The formula for the area of a rectangle is length multiplied by width. In this case, the length is "l" and the width is the shorter side of the rectangle, which we'll call "w".

So, we have l * w = 0.6s^2.

From our previous equation, we know that s = l, so we can substitute s for l in the equation above:

l * w = 0.6l^2

Dividing both sides of the equation by l, we get:

w = 0.6l

Now we have an expression for the width (shorter side) of the rectangle in terms of the longer side (l), which will help us find the ratio of the shorter side to the longer side.

To find the ratio of the shorter side to the longer side, we divide the width (w) by the length (l):

w/l = (0.6l)/l = 0.6

Therefore, the ratio of the shorter side to the longer side of the rectangle is 0.6.