Question

quadrilateral ( a ^ { prime } b ^ { prime } c ^ { prime } d ^ { prime } ) is the result of reflecting quadrilateral ( a b c d ) across the line ( l ).
select all of the correct statements about the unchanged properties of quadrilateral ( a b c d ) and quadrilateral ( a ^ { prime } b ^ { prime } c ^ { prime } d ^ { prime } ).
choose all answers that apply:
the perimeters of quadrilateral ( a b c d ) and quadrilateral ( a ^ { prime } b ^ { prime } c ^ { prime } d ^ { prime } ) are the same.
( a d ) and ( a ^ { prime } d ^ { prime } ) have the same lengths.
( c ) and ( c ^ { prime } ) have the same coordinates.

Explanation:

Step1: Properties of reflection

Reflection is a rigid transformation. Rigid transformations preserve lengths and angles. So, the perimeter (sum of side lengths) of the pre - image (quadrilateral \(ABCD\)) and the image (quadrilateral \(A'B'C'D'\)) is the same. Also, corresponding sides (such as \(AD\) and \(A'D'\)) have the same lengths.
For a point \(C\) and its image \(C'\) under reflection across a line \(l\), unless the point \(C\) lies on the line of reflection \(l\), \(C\) and \(C'\) do not have the same coordinates. In a general reflection (not when the point is on the line of reflection), the coordinates of \(C\) and \(C'\) are different.

Answer:

A. The perimeters of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are the same.
B. \(AD\) and \(A'D'\) have the same lengths.