To determine which equation Alicia could use to show that the two quadrilaterals \( PQRS \) and \( TUVW \) are congruent, we need to consider the types of transformations involved in each equation.
- The equation involves operations like rotation \( r \) and translation \( T \).
- For two shapes to be congruent, one shape can be transformed into the other using a combination of rigid motions (which include rotations, translations, and reflections).
Now, let's review the provided options:
-
\(\left(r_{(90^{\circ},O)}\circ T_{<0,-3>}\right)(PQRS) = TUVW\): This represents a 90-degree rotation around point \( O \), followed by a translation of \( <0, -3> \). This could be useful, but depends on the specific positions of the points.
-
\(\left(r_{(180^{\circ},O)}\circ T_{<0,-3>}\right)(PQRS) = TUVW\): This is a 180-degree rotation around point \( O \), followed by a translation of \( <0, -3> \). This transformation can change the orientation, and if the quadrilaterals are congruent but not oriented identically, this could show that they are congruent.
-
\(\left(r_{(180^{\circ},O)}\circ T_{<-3,0>}\right)(PQRS) = TUVW\): This is similar to the previous one but involves a translation of \( <-3, 0> \). Again, whether this is valid depends on the specific coordinates of \( PQRS \) and \( TUVW \).
-
\(\left(r_{(90^{\circ},O)}\circ T_{<-3,0>}\right)(PQRS) = TUVW\): A translation of \( <-3, 0> \) followed by a 90-degree rotation around point \( O \). Like the previous transformations, the congruence would depend on the shapes' positions.
To determine which one demonstrates congruence, we must know the details concerning the positions and orientations of quadrilaterals \( PQRS \) and \( TUVW \). However, typically for congruence without specific coordinates given, the transformations that maintain congruence without affecting relative sizes or shapes—pure rotations and translations—are generally sufficient.
Among the transformations presented, the ones involving a 180-degree rotation are most effective because a 180-degree rotation changes orientation without affecting the properties of congruence (any two shapes that can be obtained from each other by a rotation and possibly a translation are congruent).
Therefore, the best option indicating congruence would likely be:
- \(\left(r_{(180^{\circ},O)}\circ T_{<0,-3>}\right)(PQRS) = TUVW\)
This option covers a rotation that could align \( PQRS \) and \( TUVW \) with the appropriate translation for their overlap or alignment. Hence, this would be my recommended choice to show that the quadrilaterals are congruent.
11 answers