To determine the location of the original vertex [tex]\( Q \)[/tex], before the transformation [tex]\( R_{0,90} \)[/tex], we need to consider the nature of the transformation rule. The rule [tex]\( R_{0,90} \)[/tex] denotes a 90-degree rotation counterclockwise around the origin. To find the original position of a vertex after such a rotation, we can instead consider applying a 90-degree clockwise rotation to the transformed vertex.
The general rule for rotating a point [tex]\((x, y)\)[/tex] 90 degrees counterclockwise is to transform it to [tex]\((-y, x)\)[/tex]. Conversely, to reverse this and rotate a point [tex]\((x', y')\)[/tex] 90 degrees clockwise to recover the original point [tex]\((x, y)\)[/tex], we use the transformation [tex]\((x', y') \to (y', -x')\)[/tex].
Given the transformed coordinates of [tex]\(Q'\)[/tex] are [tex]\((-3, 4)\)[/tex]:
1. Let the coordinates of [tex]\( Q' \)[/tex] be [tex]\((-3, 4)\)[/tex]. Here [tex]\( x' = -3 \)[/tex] and [tex]\( y' = 4 \)[/tex].
2. Applying the rule for a 90-degree clockwise rotation:
[tex]\[
x = y' = 4
\][/tex]
[tex]\[
y = -x' = -(-3) = 3
\][/tex]
Therefore, the original coordinates of [tex]\( Q \)[/tex] are [tex]\((4, 3)\)[/tex].
So, the location of [tex]\( Q \)[/tex] is:
[tex]\[
(4, 3)
\][/tex]
Thus, the correct answer is [tex]\((4, 3)\)[/tex].
