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The square's perimeter is $12\times 4 = 48$
You could also look at the area of the rectangle. This is $16 \times 9 = 144$. This must be the same as the square, so the square must have side lengths of $12$. Therefore its perimeter is $12 \times 4 = 48$.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.