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'Who is the fairest of them all ?' printed from https://nrich.maths.org/
If the flag is enlarged by scale factor $k$ and then by scale
factor $\frac{1}{k}$, how large will it be at the end? Will it have
changed the way it is facing?
You might find it useful to use vectors for the proof at the end of
this question. For example, you could write $\mathbf{x}$ for the
vector from the first centre of enlargement to the second centre of
enlargement, and $\mathbf{a}$ for the vector from the first centre
of enlargement to the foot of the flagpole. If you can prove your
answer for the foot of the flagpole, do you need to do any extra
work to prove it for the rest of the flag?