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'Golden Construction' printed from https://nrich.maths.org/
Once you have done the construction the different parts of this
investigation can be taken separately and there is no need to
tackle every part.
The problem solver is led through several different ways of finding
the value of the Golden Ratio $\phi$ and proving that the various
relationships are equivalent.
Younger problem solvers can simply stop at measuring the lengths
and finding that the ratio of the sides of the rectangle $AEFD$ is
the same as the ratio of the sides of the smaller rectangle $CBEF$
you get by removing a square from $AEFD$. This shows that $AEFD$
and $CBEF$ are golden rectangles.
You need Pythagoras' theorem and to be able to work with surds to
get an exact value of this ratio in terms of $\sqrt 5$.
Alternatively, if you call this ratio $\phi$, then you can deduce
that $$\phi = 1 + {1\over \phi}$$ and use a graphical method to
solve this equation.
Lastly, if you know about quadratic equations, you can solve the
quadratic equation to find the value of $\phi$.