Question

the image of trapezoid pqrs after a reflection across line wy is trapezoid pqrs.
what is the relationship between segment rr and segment ss?
options:

• rr = 2ss
• rr = 4ss
• segment rr is parallel to segment ss
• segment rr is perpendicular to segment ss

Explanation:

Step1: Recall Reflection Properties

In a reflection across a line, the segment joining a point and its image (e.g., \( \overline{RR'} \) and \( \overline{SS'} \)) is perpendicular to the line of reflection, and the line of reflection is the perpendicular bisector of these segments. Also, the segments \( \overline{RR'} \) and \( \overline{SS'} \) are both perpendicular to the same line (the line of reflection \( \overleftrightarrow{WY} \)), so they are parallel to each other.

Step2: Analyze Options

• \( RR' = 2SS' \): Not true, as reflection preserves distance from the line, so \( RR' \) and \( SS' \) should be equal in length (or related by the same scale, but here they are both perpendicular to the same line, so parallel, not necessarily in a 2:1 ratio).
• \( RR' = 4SS' \): Similar to above, no basis for a 4:1 ratio.
• \( \overline{RR'} \parallel \overline{SS'} \): Since both are perpendicular to the line of reflection \( \overleftrightarrow{WY} \), by the theorem that two lines perpendicular to the same line are parallel, this is true.
• \( \overline{RR'} \perp \overline{SS'} \): If both are perpendicular to the same line, they can't be perpendicular to each other (they are parallel).

Answer:

\( \overline{RR'} \parallel \overline{SS'} \) (the option with this statement)