A rectangle has width 1 unit and length x units.

A square is cut off it.

the rectangle left behind is similar to the original rectangle.

Find the value of x in surd form.

The value of x is called Golden ratio

MY attempt:

area of the original rectangle = x sq units

area of square = 1 sq units

are of b ( small rectangle) = x-1 sq units

i found the golden ratio by doing this:

$\displaystyle x = \frac{1}{x-1} $

$\displaystyle x(x-1)=1 $

$\displaystyle x^2 -x=1 $

$\displaystyle (x-\frac{1}{2})^2-\frac{1}{4}=1 $

$\displaystyle (x-\frac{1}{2})^2 = \frac{5}{4} $

$\displaystyle x = \frac{1}{2}\pm\frac{\sqrt{5}}{2} $

since width and length are positive units

thus $\displaystyle x = \frac{1}{2} + \frac{\sqrt{5}}{2} $

I found the answer but I read off a forum that i needed to set up the equations relating ratios involving x

but when i tihnk about it why is $\displaystyle x = \frac{1}{x-1} $ this is the part im confused about

in my head 3 = (6/2) thus i thought for a long time that the ratio should be $\displaystyle 1 = \frac{x}{x-1} $ not $\displaystyle x = \frac{1}{x-1} $

Pleqase can some explain more in order to deepen my understanding thank you