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Which series of transformations can be used to show that PQRS is similar to P'Q'R'S' is it a 90° clockwise rotation about R followed by a dilation through R by scale factor PQ/P'Q' or P'Q'/PQ I'm getting two different answers
Which series of transformations can be used to show that PQRS is similar to P'Q'R'S' is it a 90° clockwise rotation about R followed by a dilation through R by scale factor PQ/P'Q' or P'Q'/PQ I'm getting two different answers
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Answer
PQRS is smaller than P'Q'R'S' does that mean it should be P'Q'/PQ
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GPT-4o mini
To determine which series of transformations can show that quadrilateral \( PQRS \) is similar to quadrilateral \( P'Q'R'S' \), we need to consider both rotation and dilation.
1. **Rotation**:
A 90° clockwise rotation about point \( R \) will change the positions of points \( P \), \( Q \), and \( S \) around point \( R \). We need to check if this rotation aligns \( PQRS \) with \( P'Q'R'S' \) properly.
2. **Dilation**:
After rotating, we would then apply a dilation centered at point \( R \). The scale factor can be determined by the lengths of corresponding sides. If \( PQ \) corresponds to \( P'Q' \), then the scale factor can be calculated as either:
- \( \text{Scale Factor} = \frac{PQ}{P'Q'} \) (if \( P'Q' \) is an enlargement of \( PQ \))
- or \( \text{Scale Factor} = \frac{P'Q'}{PQ} \) (if \( P'Q' \) is a reduction of \( PQ \))
To confirm the order of transformations:
- First, apply the 90° clockwise rotation about \( R \) to obtain a new configuration of \( PQRS \).
- Then, apply the dilation through \( R \) using the appropriate scale factor determined from the corresponding lengths.
Since you're unsure about the correct scale factor, it largely depends on whether the transformation from \( PQRS \) to \( P'Q'R'S' \) involves an enlargement (using \( PQ/P'Q' \)) or a reduction (using \( P'Q'/PQ \)).
To summarize, the correct series of transformations would be:
1. Perform a **90° clockwise rotation about point \( R \)**.
2. Follow with a **dilation from point \( R \)**, using the correct scale factor based on the relationships between corresponding sides (either \( PQ/P'Q' \) if it's a reduction or \( P'Q'/PQ \) if it's an enlargement).
To determine whether to use \( PQ/P'Q' \) or \( P'Q'/PQ \), you'll need to analyze the relationship of the corresponding sides of the two quadrilaterals.
1. **Rotation**:
A 90° clockwise rotation about point \( R \) will change the positions of points \( P \), \( Q \), and \( S \) around point \( R \). We need to check if this rotation aligns \( PQRS \) with \( P'Q'R'S' \) properly.
2. **Dilation**:
After rotating, we would then apply a dilation centered at point \( R \). The scale factor can be determined by the lengths of corresponding sides. If \( PQ \) corresponds to \( P'Q' \), then the scale factor can be calculated as either:
- \( \text{Scale Factor} = \frac{PQ}{P'Q'} \) (if \( P'Q' \) is an enlargement of \( PQ \))
- or \( \text{Scale Factor} = \frac{P'Q'}{PQ} \) (if \( P'Q' \) is a reduction of \( PQ \))
To confirm the order of transformations:
- First, apply the 90° clockwise rotation about \( R \) to obtain a new configuration of \( PQRS \).
- Then, apply the dilation through \( R \) using the appropriate scale factor determined from the corresponding lengths.
Since you're unsure about the correct scale factor, it largely depends on whether the transformation from \( PQRS \) to \( P'Q'R'S' \) involves an enlargement (using \( PQ/P'Q' \)) or a reduction (using \( P'Q'/PQ \)).
To summarize, the correct series of transformations would be:
1. Perform a **90° clockwise rotation about point \( R \)**.
2. Follow with a **dilation from point \( R \)**, using the correct scale factor based on the relationships between corresponding sides (either \( PQ/P'Q' \) if it's a reduction or \( P'Q'/PQ \) if it's an enlargement).
To determine whether to use \( PQ/P'Q' \) or \( P'Q'/PQ \), you'll need to analyze the relationship of the corresponding sides of the two quadrilaterals.
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