1. rotate the figure $180^{\\circ}$ counterclockwise about the origin

Explanation:

Step1: Identify the coordinates of the vertices

From the grid, the coordinates are $A(-4, 2)$, $B(-1, 5)$, and $C(-2, 1)$.

Step2: Apply the rotation rule

A $180^{\circ}$ rotation about the origin uses the rule $(x, y) \rightarrow (-x, -y)$.

Step3: Calculate the new coordinates

For $A(-4, 2)$: $A'(-(-4), -(2)) = (4, -2)$. For $B(-1, 5)$: $B'(-(-1), -(5)) = (1, -5)$. For $C(-2, 1)$: $C'(-(-2), -(1)) = (2, -1)$.

Answer:

The rotated figure has vertices at $A'(4, -2)$, $B'(1, -5)$, and $C'(2, -1)$.