To identify a sequence of rigid transformations that would map quadrilateral ABCD to A"B"C"D", we need to analyze what each transformation does.
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Option a: (x,y) -> (-x - 4, y)
- This transformation reflects the point over the y-axis and then shifts it left by 4 units.
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Option b: (x,y) -> (-x, y - 4)
- This transformation reflects the point over the y-axis and then shifts it down by 4 units.
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Option c: (x,y) -> (x, y + 4)
- This transformation shifts the point up by 4 units.
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Option d: (x,y) -> (x, -y - 4)
- This transformation reflects the point over the x-axis and then shifts it down by 4 units.
Now, let's construct a composition of transformations that will map ABCD to A"B"C"D".
Based on the nature of transformations, we can start with a reflection and translations.
To outline an example sequence:
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Apply transformation d: (x,y) -> (x, -y - 4)
This would reflect the points over the x-axis and shift them down. -
Follow it with transformation c: (x,y) -> (x, y + 4)
This shifts the resulting points up by 4.
Thus, the resulting composition of transformations will be:
\[ (x,y) \xrightarrow{d} (x, -y - 4) \xrightarrow{c} (x, -(-y - 4) + 4) = (x, y) \]
However, to map ABCD specifically to A"B"C"D" may require a precise arrangement or specific parameters around the specific shape and placement of points.
To codify the transformations in our required mapping:
- First use option d.
- Use option c afterwards.
So you would write the final composition transformation as: d followed by c:
\[ (x,y) \xrightarrow{d} (x, -y - 4) \xrightarrow{c} (x, -y - 4 + 4) = (x, -y) \]
This means: Final composition: Use the transformations in the sequence as \[ d \circ c \].
In a clean answer format, you'd express it this way:
The sequence of transformations is: \( d \) followed by \( c \) (i.e., \( d \circ c \)).
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