Question

1. which sequence of transformations will carry the given pre - image onto the image shown with dashed lines?

Explanation:

To solve this, we analyze the transformation:

Step 1: Identify Rotation

First, rotate the pre - image (rectangle \(ABCD\)) 90 degrees clockwise (or 270 degrees counter - clockwise) about the origin (or a suitable center). A 90 - degree clockwise rotation of a point \((x,y)\) gives \((y, - x)\). This rotation will change the orientation of the rectangle to be horizontal (matching the general orientation of the dashed - line image).

Step 2: Identify Translation

After the rotation, we need to translate the rotated figure so that it aligns with the dashed - line image ( \(A'B'C'D'\) ). The translation will involve moving the figure horizontally and/or vertically to match the position of the image. For example, if after rotation the figure is in a certain position, we can translate it left/right and up/down to get it to the exact position of the dashed - line image.

(Note: Since the exact coordinates are not given, we rely on the visual change in orientation (from vertical to horizontal, indicating a 90 - degree rotation) and then a translation to match the position. The key sequence is a 90 - degree clockwise rotation followed by a translation (or other valid combinations depending on the exact positions, but rotation to change orientation is a key first step here).)

If we assume the standard approach for such rectangle transformation problems:

1. Rotation: 90° clockwise about the origin (or a vertex).
2. Translation: Move the rotated figure to the position of the dashed - line image.

(If this was a multiple - choice question with options like "90° clockwise rotation and then a translation", that would be the answer. Since the final answer format is not fully specified but the problem is about geometric transformations in Geometry (subfield of Mathematics), the sequence involves a rotation (90° clockwise) and a translation. )

Answer:

A possible sequence is a 90° clockwise rotation about the origin (or a suitable center) followed by a translation to align with the dashed - line image. (If in a multiple - choice context, select the option with 90° clockwise rotation and translation.)