To determine the transformation that maps triangle \( P \) to triangle \( R \) after reflecting \( P \) across line \( l \) to create triangle \( Q \) and then reflecting \( Q \) across line \( m \), we can analyze the effects of these two reflections.
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Reflection across line \( l \): This first reflection will create a triangle \( Q \) that is a mirror image of triangle \( P \) with respect to line \( l \).
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Reflection across line \( m \): The second reflection will take triangle \( Q \) and reflect it across line \( m \), which is parallel to line \( l \).
When reflecting across two parallel lines \( l \) and \( m \), the overall effect is equivalent to a translation. The distance between lines \( l \) and \( m \) doubles the distance of any points in triangle \( P \) to their images in triangle \( R \).
Thus, the transformation that maps triangle \( P \) to triangle \( R \) is best described as a translation (specifically, a translation in the direction which would take points in \( P \) to points in \( R \)). Since a reflection across two parallel lines is a translation, the answer is:
A. a translation to the right (or potentially B. a translation to the left, depending on the specific orientation of the lines and the triangles—though typically it is understood to go in a consistent direction based on the stated problem).
However, without additional context regarding the specific orientation and position of triangles \( P \), \( Q \), and \( R \), it is assumed to be a single consistent direction based on the layout of the figure.