To determine which equation could be used to show that the two quadrilaterals \(PQRS\) and \(TUVW\) are congruent, we need to focus on the sequence of transformations involved in each option.
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Translation followed by Rotation: \(T_{<0,-3>}\) would move the shape vertically down and \(r_{(90^\circ,O)}\) would rotate it 90 degrees around the origin.
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Translation followed by Rotation: \(T_{<0,-3>}\) followed by \(r_{(180^\circ,O)}\) would translate down and then rotate it 180 degrees around the origin.
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Translation followed by Rotation: \(T_{<-3,0>}\) then followed by \(r_{(180^\circ,O)}\) translates the shape left and then rotates it 180 degrees.
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Translation followed by Rotation: \(T_{<-3,0>}\) followed by \(r_{(90^\circ,O)}\) translates the shape left and then rotates it 90 degrees.
To determine which of these transformations shows congruence, we need to analyze the provided transformations in terms of possible congruity:
- Congruent shapes maintain their size and shape regardless of translation or rotation.
- Rotating (180 degrees) or (90 degrees) along with the right translations may align and create congruence.
Without specific coordinates of the quadrilaterals, we cannot definitively determine the correct transformations. However, generally, if one shape can be aligned and adjusted to match the other through rigid transformations (translations and rotations) without distortion, then they are congruent.
Based on typical problems of this nature, often a \(180^\circ\) rotation tends to preserve the properties of congruence effectively. Thus, the second option seems more likely to prove congruency since it uses a \(180^\circ\) rotation that is commonly found in congruence conditions:
\[ \large \left(r_{\left(180^{\circ},O\right)}\circ T_{<0,-3>}\right)\left(PQRS\right)=TUVW \]
This indicates that it rotations and translates the shape, likely resulting in congruent figures.
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