1. Let's consider the example of a rotation around the origin by an angle $\theta = \frac{\pi}{4}$ (45 degrees) counter-clockwise. 2. The rotation transformation is given by the formula: $$r_{O,\theta}(x,y) = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)$$ 3. For $\theta = \frac{\pi}{4}$, we have $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ and $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$. 4. Applying the rotation to a point, for example $(1,0)$: $$r_{O,\frac{\pi}{4}}(1,0) = (1 \cdot \frac{\sqrt{2}}{2} - 0 \cdot \frac{\sqrt{2}}{2}, 1 \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2}) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$ 5. This means the point $(1,0)$ is rotated 45 degrees counter-clockwise to the point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. 6. This transformation preserves distances and angles, which is a key property of Euclidean isometries. 7. The illustration would show the original point $(1,0)$ on the x-axis and the rotated point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ on the line $y=x$, both at the same distance from the origin.