The problem asks us to graph the image of triangle $LMN$ after a 180-degree counterclockwise rotation around the origin. From the graph, the coordinates of the vertices of triangle $LMN$ are approximately $L(-4, 7)$, $M(1, 7)$, and $N(-4, 6)$.

1. Problem Description

The problem asks us to graph the image of triangle $LMN$ after a 180-degree counterclockwise rotation around the origin. From the graph, the coordinates of the vertices of triangle $LMN$ are approximately $L(-4, 7)$, $M(1, 7)$, and $N(-4, 6)$.

2. Solution Steps

A 180-degree rotation around the origin transforms a point $(x, y)$ to $(-x, -y)$. Therefore, we apply this transformation to the coordinates of $L$, $M$, and $N$.

The coordinates of $L'$ are $(-(-4), -7) = (4, -7)$.

The coordinates of $M'$ are $(-1, -7)$.

The coordinates of $N'$ are $(-(-4), -6) = (4, -6)$.

3. Final Answer

The coordinates of the vertices of the rotated triangle $L'M'N'$ are $L'(4, -7)$, $M'(-1, -7)$, and $N'(4, -6)$.

(Note: The solution only provides the coordinates. To fully answer the question, you would need to plot these points on a coordinate plane and connect them to form the triangle.)