Paper 3, Section I, D

Show that every orthogonal $2 \times 2$ matrix $R$ is the product of at most two reflections in lines through the origin.

Every isometry of the Euclidean plane $\mathbb{R}^{2}$ can be written as the composition of an orthogonal matrix and a translation. Deduce from this that every isometry of the Euclidean plane $\mathbb{R}^{2}$ is a product of reflections.

Give an example of an isometry of $\mathbb{R}^{2}$ that is not the product of fewer than three reflections. Justify your answer.

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