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'Golden Construction' printed from https://nrich.maths.org/
Just do part 1 if you like where you will construct a golden
rectangle and find an approximate value of the golden ratio by
measurement.
If you want to go further there are many hints within this question
to take you through it. You will find the value of $\phi$ first by
drawing and measurement, then by using geometry and calculating
with surds (square roots), then by taking a reading from a graph
and finally by rearranging a formula into a quadratic equation and
solving it.