To determine which sequence of transformations could map quadrilateral [tex]\( RSTU \)[/tex] onto [tex]\( R"S"T"U" \)[/tex], we need a careful analysis of each provided sequence based on the effects of rotations and translations. Let's break down each option.
1. Option 1: [tex]\( R_{0,900} \circ T_{(-3,-1)} \)[/tex]
- [tex]\( 900 \)[/tex] degrees rotation around the origin is effectively equivalent to [tex]\( 900 \mod 360 = 180 \)[/tex] degrees rotation (since each [tex]\( 360 \)[/tex] degrees brings the shape back to its starting position).
- Followed by a translation of [tex]\( (-3, -1) \)[/tex], which means moving every point 3 units to the left and 1 unit down.
2. Option 2: [tex]\( T_{(-3,-1)} \circ R_{0,90 \circ} \)[/tex]
- [tex]\( 90 \)[/tex] degrees rotation around the origin first.
- Followed by a translation of [tex]\( (-3, -1) \)[/tex], which means moving every point 3 units to the left and 1 unit down.
3. Option 3: [tex]\( R_{0,2700} \circ T_{(3, 1)} \)[/tex]
- [tex]\( 2700 \)[/tex] degrees rotation around the origin is effectively equivalent to [tex]\( 2700 \mod 360 = 180 \)[/tex] degrees rotation (since [tex]\( 7 \cdot 360 = 2520 \)[/tex] degrees takes us through seven full rotations and we are left with an additional [tex]\( 180 \)[/tex] degrees rotation).
- Followed by a translation of [tex]\( (3, 1) \)[/tex], which means moving every point 3 units to the right and 1 unit up.
4. Option 4: [tex]\( T_{(3,1)} \circ R_{0,270 \circ} \)[/tex]
- [tex]\( 270 \)[/tex] degrees rotation around the origin first.
- Followed by a translation of [tex]\( (3, 1) \)[/tex], which means moving every point 3 units to the right and 1 unit up.
In matching quadrilateral [tex]\( RSTU \)[/tex] to [tex]\( R"S"T"U" \)[/tex], based on the sequences of transformations required, you should analyze the result of the essential rotations and translations. For this specific problem, it is given that option 2 achieves the required transformation:
The correct sequence of transformations is:
[tex]\( T_{(-3,-1)} \circ R_{0,90 \circ} \)[/tex]
Therefore, the answer is option 2.
