To find the location of [tex]\( Q^2 \)[/tex], you need to understand the transformation rule [tex]\( R_{0,90^{\circ}} \)[/tex], which is a 90-degree counterclockwise rotation around the origin. Specifically, when you apply this rule to a point [tex]\((x, y)\)[/tex], the resulting point becomes [tex]\((-y, x)\)[/tex].
Here, you are given the transformed coordinates of the points of a rectangle. To find the original coordinates of the point [tex]\( Q \)[/tex] (before the transformation that rotated it to [tex]\( Q^{\prime} \)[/tex]), you must reverse the rotation transformation.
Given [tex]\( Q^{\prime}(-3, 4) \)[/tex], we consider the inverse of the [tex]\( R_{0,90^{\circ}} \)[/tex] transformation. When reversing this transformation, which rotated points 90 degrees counterclockwise to get to their current coordinates, we rotate them 90 degrees clockwise to determine their starting coordinates. The inverse relation for a 90-degree clockwise rotation is [tex]\((x', y') \rightarrow (y', -x')\)[/tex].
Applying this inverse transformation:
[tex]\[ Q'(-3, 4) \][/tex]
So, we have:
[tex]\[ x' = -3 \][/tex]
[tex]\[ y' = 4 \][/tex]
To find the original location [tex]\( Q^2 \)[/tex], we apply the inverse transformation rule:
[tex]\[ Q^2 = (y', -x') \][/tex]
Therefore:
[tex]\[ Q^2 = (4, 3) \][/tex]
So, the location of [tex]\( Q^2 \)[/tex] is [tex]\( (4, 3) \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{(4, 3)} \][/tex]
