To rotate a point \((x, y)\) counterclockwise about the origin by 270 degrees, we can use the rotation transformation formulas. The general formula for rotating a point \((x, y)\) by an angle \(\theta\) counterclockwise is given by:
\[ (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \]
For a rotation of 270 degrees (or \(\frac{3\pi}{2}\) radians), the cosine and sine values are:
\[ \cos(270^\circ) = 0 \] \[ \sin(270^\circ) = -1 \]
Now substituting these values into the rotation formulas:
\[ x' = x \cdot 0 - y \cdot (-1) = y \] \[ y' = x \cdot (-1) + y \cdot 0 = -x \]
Thus, after rotating the point \((x, y)\) counterclockwise by 270 degrees, the new coordinates of the point become:
\[ (y, -x) \]
In conclusion, the new point after a 270-degree counterclockwise rotation is \((y, -x)\).
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