Questions: The point C(4,5) is rotated 270° counterclockwise around the origin. What are the coordinates of the resulting point, C'?

Solution Steps

To rotate a point \((x, y)\) counterclockwise around the origin by an angle \(\theta\), the new coordinates \((x', y')\) are given by: \[ x' = x \cos \theta - y \sin \theta \] \[ y' = x \sin \theta + y \cos \theta \]

The point \(C(4, 5)\) is rotated \(270^\circ\) counterclockwise. We substitute \(x = 4\), \(y = 5\), and \(\theta = 270^\circ\) into the rotation formulas.

First, calculate \(\cos 270^\circ\) and \(\sin 270^\circ\): \[ \cos 270^\circ = 0 \] \[ \sin 270^\circ = -1 \]

Substitute the values into the rotation formulas: \[ x' = 4 \cdot 0 - 5 \cdot (-1) = 0 + 5 = 5 \] \[ y' = 4 \cdot (-1) + 5 \cdot 0 = -4 + 0 = -4 \]

Final Answer