Subjects geometry

90 Degree Rotation

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1. **Stating the problem:** We need to find the result of a 90° counterclockwise rotation about the origin. 2. **Formula for rotation:** A rotation of a point $(x,y)$ about the origin by an angle $\theta$ counterclockwise is given by: $$ (x', y') = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta) $$ 3. **Apply the formula for 90° counterclockwise rotation:** Here, $\theta = 90^\circ = \frac{\pi}{2}$ radians. We know: $$ \cos 90^\circ = 0, \quad \sin 90^\circ = 1 $$ So, $$ (x', y') = (x \cdot 0 - y \cdot 1, x \cdot 1 + y \cdot 0) = (-y, x) $$ 4. **Interpretation:** This means every point $(x,y)$ after a 90° counterclockwise rotation about the origin moves to $(-y, x)$. **Final answer:** The transformation for a 90° counterclockwise rotation about the origin is: $$ (x,y) \to (-y, x) $$