Question

the image of trapezoid pqrs after a reflection across $overleftrightarrow{wy}$ is trapezoid pqrs. what is the relationship between $overline{rr}$ and $overline{ss}$? $\bigcirc$ $rr = 2ss$ $\bigcirc$ $rr = 4ss$ $\bigcirc$ $overline{rr} parallel overline{ss}$ $\bigcirc$ $overline{rr} perp overline{ss}$

Explanation:

Step1: Recall Reflection Properties

In a reflection across a line (here, \(\overleftrightarrow{WY}\)), the line of reflection is the perpendicular bisector of the segment joining a point and its image (e.g., \(RR'\) and \(SS'\) are segments joining \(R\) to \(R'\) and \(S\) to \(S'\), respectively). Also, all such segments (from original points to their images) are parallel to each other because they are all perpendicular to the line of reflection.

Step2: Analyze Each Option

• Option 1: \(RR' = 2SS'\) – There's no reason for \(RR'\) to be twice \(SS'\); reflection preserves distance, so \(RR' = SS'\) (lengths of segments from a point to its image are equal), so this is wrong.
• Option 2: \(RR' = 4SS'\) – Similar to above, lengths should be equal, so this is wrong.
• Option 3: \(\overline{RR'} \parallel \overline{SS'}\) – Since both \(RR'\) and \(SS'\) are perpendicular to the line of reflection \(\overleftrightarrow{WY}\), two lines perpendicular to the same line are parallel. So this is correct.
• Option 4: \(\overline{RR'} \perp \overline{SS'}\) – If both are perpendicular to \(\overleftrightarrow{WY}\), they can't be perpendicular to each other (they are parallel), so this is wrong.

Answer:

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