The problem asks us to find the image of the point $(5, 0)$ after a rotation of $90^\circ$ counterclockwise about the origin.

1. Problem Description

The problem asks us to find the image of the point $(5, 0)$ after a rotation of $90^\circ$ counterclockwise about the origin.

2. Solution Steps

A rotation of $90^\circ$ counterclockwise about the origin transforms a point $(x, y)$ to $(-y, x)$. This is because the rotation matrix for a $90^\circ$ counterclockwise rotation is given by:

$\begin{bmatrix} \cos(90^\circ) & -\sin(90^\circ) \\ \sin(90^\circ) & \cos(90^\circ) \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

So, if we have a point $(x, y)$, we multiply the matrix by the column vector $\begin{bmatrix} x \\ y \end{bmatrix}$ to obtain:

$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -y \\ x \end{bmatrix}$

Thus, the new coordinates are $(-y, x)$.

For the point $(5, 0)$, we have $x = 5$ and $y = 0$. Applying the rotation rule, the image of the point $(5, 0)$ is $(-0, 5)$, which simplifies to $(0, 5)$.

3. Final Answer

The image of the point $(5, 0)$ after a $90^\circ$ counterclockwise rotation is $(0, 5)$. Therefore, the correct answer is C. $(0, 5)$.