point A is the original point before rotating. Which point represents A' after A (4,1) is rotated 90 degrees counterclockwise: B, C, D, E, F, G, or H ?
To rotate a point counterclockwise about the origin, we can use the following formula:\[\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix},\]where $(x, y)$ represents the original point, $(x', y')$ represents the rotated point, and $\theta$ represents the angle of rotation.
Since we are rotating point A (4,1) counterclockwise by 90 degrees, we have:\[\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos(90^\circ) & -\sin(90^\circ) \\ \sin(90^\circ) & \cos(90^\circ) \end{pmatrix} \begin{pmatrix} 4 \\ 1 \end{pmatrix}.\]Using the values of $\cos(90^\circ) = 0$ and $\sin(90^\circ) = 1,$ the transformation becomes:\[\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 4 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}.\]This rotated point is represented by $\boxed{E}.$