In the given square grid below,

1. Determine the grid vector for line segment AB.

Let the coordinates of A be $(x_A, y_A)$ and B be $(x_B, y_B)$. The vector $\vec{AB}$ is given by the horizontal and vertical grid steps from A to B. Suppose $\vec{AB} = \begin{pmatrix} p \\ q \end{pmatrix}$.

2. Determine the grid vector for AE to satisfy $\angle BAE = 90^\circ$ and $AB = AE$.

For $\angle BAE = 90^\circ$ and $AB = AE$, the vector $\vec{AE}$ must be a $90^\circ$ rotation of $\vec{AB}$. A $90^\circ$ rotation of $\begin{pmatrix} p \\ q \end{pmatrix}$ gives either $\begin{pmatrix} -q \\ p \end{pmatrix}$ or $\begin{pmatrix} q \\ -p \end{pmatrix}$. Both preserve the length (since $\sqrt{(-q)^2 + p^2} = \sqrt{p^2 + q^2}$) and are perpendicular to $\vec{AB}$.

3. Locate point E and draw triangle ABE.

From point A, move according to the vector $\vec{AE}$ to locate point E. Draw the line segments AE and BE to complete the triangle ABE.