Golden ratio

Golden boxes
if you take a rectangle, and you remove a square from one end of it, which has sides equal to the length of the shorter side of the rectangle. If the rectangle that remains has the same ratio of each of its sides to the original rectangle. What is the length of sides (or ratio of sides), to the original rectangle?

GoogleDrawing/1dKxqvaHsqYdDxoTDru6wExxrUZLIGlot7myR95SqOSk/960/720

so we can annotate the drawing with algebraic representations of the relative proportions

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So the intention is to solve the problem of finding the smaller rectangle with the similar shape, and hence the same ratio as the larger rectangle. hence

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

or with the longer side is the numerator

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

hence

$x(x-1) = 1$

$x^2-x-1=0$

as $\{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\}$

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

$\approx 1.618033989$

Golden Lines
Can a line be divided in such a way, that the ratio of AP to AB is the same as the ratio of PB to AP

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Properties of the golden numbers
$\phi = \frac{1}{2}(1+\sqrt{5}) = 1.618.....$

$\phi = \frac{1}{2}(1+\sqrt{5}) = 1.618.....$