Given the vertices of polygon [tex]\( KLMN \)[/tex] with the initial coordinates of point [tex]\( N \)[/tex] being [tex]\( (-4, 2) \)[/tex], we need to determine the new coordinates of this point after a [tex]\( 90^\circ \)[/tex] counterclockwise rotation.
When a point [tex]\((x, y)\)[/tex] is rotated [tex]\( 90^\circ \)[/tex] counterclockwise around the origin, the new coordinates [tex]\((x', y')\)[/tex] are given by the transformation:
[tex]\[ x' = -y \][/tex]
[tex]\[ y' = x \][/tex]
Now, let's apply this transformation to point [tex]\( N \)[/tex] with coordinates [tex]\((-4, 2)\)[/tex]:
1. Calculate the new [tex]\( x' \)[/tex] coordinate:
[tex]\[
x' = -y = -2
\][/tex]
2. Calculate the new [tex]\( y' \)[/tex] coordinate:
[tex]\[
y' = x = -4
\][/tex]
Thus, the new coordinates of point [tex]\( N \)[/tex] after a [tex]\( 90^\circ \)[/tex] counterclockwise rotation are [tex]\((-2, -4)\)[/tex].
Therefore, the image of [tex]\( N(-4,2) \)[/tex] after the rotation is [tex]\( N'(-2,-4) \)[/tex].
