A rectangle is transformed according to the rule [tex]\( R_{0,90^{\circ}} \)[/tex]. The image of the rectangle has vertices located at [tex]\( R'(-4,4) \)[/tex], [tex]\( S'(-4,1) \)[/tex], [tex]\( P'(-3,1) \)[/tex], and [tex]\( Q'(-3,4) \)[/tex]. What is the location of [tex]\( Q \)[/tex]?

A. [tex]\((-4, -3)\)[/tex]
B. [tex]\((-3, -4)\)[/tex]
C. [tex]\((3, 4)\)[/tex]
D. [tex]\((4, 3)\)[/tex]



Answer :

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].