Question

kenneth is creating a chart to compare and contrast the different transformations of objects. he wants to show similarities and differences between translations, rotations, and reflections. in his chart, four statements are given. which statements are true for a reflection? the distances between points on the polygon are preserved by the transformation. the orientation of points on the polygon is preserved by the transformation. the distance between corresponding points in the image and pre - image is constant. the angle measures between sides on the polygon are preserved by the transformation.

Explanation:

Step1: Recall Reflection Properties

Reflection is a rigid transformation (isometry), so it preserves distances and angle measures. Orientation (order of points, clockwise/counter - clockwise) is reversed in reflection. Also, the distance from a point to the line of reflection is equal for the pre - image and image, but the distance between corresponding points (pre - image and image) is not constant (it depends on the distance from the line of reflection), while the distance between points on the polygon (pre - image) is preserved.

Step2: Analyze Each Statement

• Statement 1: "The distances between points on the polygon are preserved by the transformation." Since reflection is a rigid transformation, distances between points on the original polygon (pre - image) are preserved. This is true.
• Statement 2: "The orientation of points on the polygon is preserved by the transformation." Reflection reverses orientation (e.g., a clockwise - ordered polygon becomes counter - clockwise after reflection). So this is false.
• Statement 3: "The distance between corresponding points in the image and pre - image is constant." The distance between a point and its image under reflection is \(2d\), where \(d\) is the distance from the point to the line of reflection. This distance varies for different points, so it's not constant. This is false.
• Statement 4: "The angle measures between sides on the polygon are preserved by the transformation." Since reflection is a rigid transformation, angle measures are preserved. This is true.

Answer:

The true statements for a reflection are:

• The distances between points on the polygon are preserved by the transformation.
• The angle measures between sides on the polygon are preserved by the transformation.