To determine the original coordinates [tex]\( Q \)[/tex] of the rectangle before it was transformed by the rule [tex]\( R_{0,90^{\circ}} \)[/tex], follow these steps for reversing a 90-degree counter-clockwise rotation.
1. Understand the given transformation:
A [tex]\( 90^{\circ} \)[/tex] counter-clockwise rotation transforms a point [tex]\((x, y)\)[/tex] into [tex]\((y, -x)\)[/tex]. To reverse this transformation, you need to find the original coordinates from the transformed ones.
2. Identify the transformed coordinates of [tex]\( Q' \)[/tex]:
The given coordinates of [tex]\( Q' \)[/tex] after transformation are [tex]\( (-3, 4) \)[/tex].
3. Apply the reverse transformation rule:
To find the original coordinates [tex]\( (x, y) \)[/tex] from [tex]\((x', y')\)[/tex] after a [tex]\( 90^{\circ} \)[/tex] counter-clockwise rotation, the rule is given by:
[tex]\[
(x, y) = (y', -x')
\][/tex]
4. Substitute the values of [tex]\( Q' \)[/tex]:
Since [tex]\( Q' \)[/tex] = [tex]\( (-3, 4) \)[/tex], you substitute these values into the rule:
[tex]\[
x = y' = 4
\][/tex]
[tex]\[
y = -x' = -(-3) = 3
\][/tex]
5. Combine the results:
So, the original coordinates [tex]\( Q \)[/tex] are:
[tex]\[
Q = (4, 3)
\][/tex]
Therefore, the location of [tex]\( Q \)[/tex] is [tex]\( (4, 3) \)[/tex].
So, the correct choice is [tex]\((4, 3)\)[/tex].
