To determine the original coordinates of [tex]\( Q \)[/tex] given its transformed coordinates [tex]\( Q'(-3, 4) \)[/tex] under the rule [tex]\( R_{0,90^{\circ}} \)[/tex], let's carefully follow these steps:
1. Understand the Transformation Rule:
The rule [tex]\( R_{0,90^{\circ}} \)[/tex] indicates a rotation of 90 degrees counterclockwise around the origin.
2. Inverse Transformation:
To find the original coordinates before the rotation, we perform the inverse of 90 degrees counterclockwise rotation, which is a 90-degree clockwise rotation.
3. Apply the Inverse Rotation:
- When a point [tex]\( (Q'_x, Q'_y) \)[/tex] is rotated 90 degrees clockwise, the new coordinates [tex]\( (Q_x, Q_y) \)[/tex] are calculated as follows:
[tex]\[
Q_x = Q'_y
\][/tex]
[tex]\[
Q_y = -Q'_x
\][/tex]
4. Given Coordinates of [tex]\( Q' \)[/tex]:
The coordinates of [tex]\( Q' \)[/tex] are [tex]\( (-3, 4) \)[/tex]. Let's substitute these values into the inverse rotation formulas:
[tex]\[
Q_x = Q'_y = 4
\][/tex]
[tex]\[
Q_y = -Q'_x = -(-3) = 3
\][/tex]
5. Determine the Coordinates:
Therefore, the coordinates of [tex]\( Q \)[/tex] are [tex]\( (4, 3) \)[/tex].
Hence, the location of [tex]\( Q \)[/tex] is [tex]\( (4, 3) \)[/tex].
The correct answer is:
[tex]\[
(4, 3)
\][/tex]
