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

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

Sagot :

GinnyAnswer
To determine the location of the vertex \(Q\) of the original rectangle given its image \( Q^{\prime} \) at \((-3, 4)\) after a \(90^\circ\) clockwise rotation around the origin, we follow these steps:

1. Understand the Transformation:
- A \(90^\circ\) clockwise rotation around the origin takes a point \((x, y)\) to \((y, -x)\).

2. Reverse the Rotation:
- To find the original coordinates of \( Q \) before the transformation, we need to reverse the \(90^\circ\) clockwise rotation.
- The reverse of a \(90^\circ\) clockwise rotation is a \(90^\circ\) counterclockwise rotation.

3. Apply the Reverse Transformation:
- For a \(90^\circ\) counterclockwise rotation, a point \((x', y')\) transforms to \((-y', x')\).

4. Calculate the Original Coordinates:
- We have \( Q^{\prime} \) at \((-3, 4)\).
- Applying the reverse transformation:
[tex]\[ x = 4 \quad \text{(formerly y')} \][/tex]
[tex]\[ y = -(-3) = 3 \quad \text{(formerly -x')} \][/tex]

Therefore, the location of \( Q \) is:
[tex]\[ \boxed{(4, 3)} \][/tex]
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