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

Step2: Analyze Each Option

• Option 1: \(RR' = 2SS'\) – No reason for this proportionality, as reflection preserves distances, so \(RR'\) and \(SS'\) should be equal in length (or at least not in a 2:1 ratio).
• Option 2: \(RR' = 4SS'\) – Similar to above, no basis for this ratio.
• Option 3: \(\overline{RR'} \parallel \overline{SS'}\) – Since both \(RR'\) and \(SS'\) are perpendicular to \(\overleftrightarrow{WY}\), they are parallel. This holds.
• 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).

Answer:

\(\boldsymbol{\overline{RR'} \parallel \overline{SS'}}\) (the option with this statement, e.g., the third option in the list)