Question

on the coordinate grid, wxyz is transformed into wxyz. choose all transformations that would prove that the two quadrilaterals are congruent. a. rotate wxyz 90° counterclockwise about the origin. b. rotate wxyz 90° counterclockwise about (-2,1). c. rotate wxyz 180° about (-5,1). d. rotate wxyz 270° clockwise about the origin. e. rotate wxyz 270° clockwise about (-2,1).

Explanation:

To determine which rotations map \( WXYZ \) to \( W'X'Y'Z' \), we analyze each option:

• Option A: Rotating 90° counterclockwise about the origin. The center of rotation for the congruence here is not the origin (the figures are shifted), so this is incorrect.
• Option B: Rotating 90° counterclockwise about \( (-2,1) \). By examining the coordinates (e.g., the center \( (-2,1) \) is a common point or midpoint of corresponding segments), this rotation aligns \( WXYZ \) with \( W'X'Y'Z' \).
• Option C: Rotating 180° about \( (-5,1) \). The center \( (-5,1) \) does not align with the symmetry of the transformation (180° rotation would require opposite points through the center, which does not match here).
• Option D: Rotating 270° clockwise about the origin. Similar to Option A, the origin is not the center of rotation for these congruent figures.
• Option E: Rotating 270° clockwise about \( (-2,1) \). A 270° clockwise rotation is equivalent to a 90° counterclockwise rotation. Since rotating 90° counterclockwise about \( (-2,1) \) (Option B) works, this also works (as 270° clockwise = 90° counterclockwise in terms of effect on the figure’s orientation relative to the center \( (-2,1) \)).

Answer:

B. Rotate \( WXYZ \) \( 90^\circ \) counterclockwise about \( (-2,1) \),
E. Rotate \( WXYZ \) \( 270^\circ \) clockwise about \( (-2,1) \)