write the coordinates of the vertices after a rotation 180° counterclockwise around the origin.

Explanation:

Step1: Recall rotation rule

The rule for a 180 - degree counter - clockwise rotation around the origin is $(x,y)\to(-x,-y)$.

Step2: Identify original coordinates

Let's assume the coordinates of points $L$, $N$, and $M$ are $L(x_1,y_1)$, $N(x_2,y_2)$, $M(x_3,y_3)$. From the graph, if $L(2, - 6)$, $N(2,-2)$, $M(8,-6)$.

Step3: Apply rotation rule to point L

For point $L(2,-6)$, after rotation, $L'(-2,6)$ since $x = 2,y=-6$ and $(-x,-y)=(-2,6)$.

Step4: Apply rotation rule to point N

For point $N(2,-2)$, after rotation, $N'(-2,2)$ since $x = 2,y = - 2$ and $(-x,-y)=(-2,2)$.

Step5: Apply rotation rule to point M

For point $M(8,-6)$, after rotation, $M'(-8,6)$ since $x = 8,y=-6$ and $(-x,-y)=(-8,6)$.

Answer:

The coordinates of the vertices after rotation are $L'(-2,6)$, $N'(-2,2)$, $M'(-8,6)$