Question

use the drop-down menus to describe the transformations used to map △abc onto △abc.

1. a rotation of dropdown counterclockwise about the origin.
2. a dilation centered at the origin with a scale factor of dropdown.

the triangles are dropdown.
(options for rotation: 90°, 180°, 270°, 360°)

Explanation:

Step1: Analyze Rotation

To map \( \triangle ABC \) to \( \triangle A'B'C' \), a rotation of \( 90^\circ \) counterclockwise about the origin would not align the triangles. A \( 180^\circ \) rotation about the origin reflects points across both axes, which aligns the orientation of \( \triangle ABC \) (with vertices in lower quadrant) to \( \triangle A'B'C' \) (upper quadrant near origin). So the rotation angle is \( 180^\circ \).

Step2: Analyze Dilation

Looking at the coordinates, original triangle \( \triangle ABC \) has larger side lengths, and \( \triangle A'B'C' \) is smaller. The scale factor for dilation (shrinking) from \( \triangle ABC \) to \( \triangle A'B'C' \) is \( \frac{1}{2} \) (since the lengths are halved, e.g., distance from \( A \) to \( B \) in original vs \( A' \) to \( B' \) in image shows scale factor \( \frac{1}{2} \)).

Answer:

1. A rotation of \( 180^\circ \) counterclockwise about the origin.
2. A dilation centered at the origin with a scale factor of \( \frac{1}{2} \).