"Golden" Rectangle

Question

Follow the instructions below and construct a rectangle. Then prove that the rectangle so constructed is a "golden" rectangle, i.e. the ratio of the longer side to the shorter side is (1+√5)/2.

Instructions: Start with a square and label the side 2x. Divide the square into 2 congruent rectangles with a vertical line. Use the diagonal of the righthand rectangle as a radius and swing an arc with center at the midpoint of the horizontal side of the square and meeting the horizontal side of the square extended. Use that intersection to construct a rectangle extending the square.

Answer 1

This is a neat (and easy) construction which has several other applications.

The diagonal of the righthand rectangle is x√5 by the Theorem of Pythagoras.

That makes the side of the constructed rectangle x(1+√5).

Then the ratio of the longer side to the shorter is (1+√5)/2