Question

which statements are true about reflections? check all that apply
an image created by a reflection will always be congruent to its pre - image
an image and its pre - image are always the same distance from the line of reflection
if a point on the pre - image lies on the line of reflection, the image of that point is the same as the pre - image
the line of reflection is perpendicular to the line segments connecting corresponding vertices
the line segments connecting corresponding vertices are all congruent to each other
the line segments connecting corresponding vertices are all parallel to each other

Explanation:

• Congruence: Reflection is a rigid transformation. Rigid transformations preserve shape and size. So, the image and pre - image are congruent.
• Distance from line of reflection: By the definition of reflection, for any point \(P\) in the pre - image and its image \(P'\), the line of reflection is the perpendicular bisector of the segment \(PP'\). So, \(P\) and \(P'\) are equidistant from the line of reflection.
• Point on line of reflection: If a point \(A\) lies on the line of reflection \(l\), then the reflection of \(A\) over \(l\) is \(A\) itself. Because the perpendicular bisector of \(AA\) (a degenerate segment with length \(0\)) is the line \(l\) (or any line passing through \(A\) in this case, but by the rule of reflection, it's mapped to itself)
• Perpendicularity: Let \(l\) be the line of reflection and \(P\) and \(P'\) be a pre - image and image pair. The reflection over line \(l\) implies that the line \(l\) is the perpendicular bisector of \(PP'\). So, the line of reflection is perpendicular to the line segment connecting corresponding vertices.

• Line segments connecting vertices:
• The line segments connecting corresponding vertices are not necessarily congruent to each other. For example, if we reflect a non - regular polygon, the lengths of these connecting segments can vary.
• The line segments connecting corresponding vertices are not necessarily parallel to each other. Consider a reflection of a polygon over a non - axis aligned line, the orientation of these connecting segments will change in a non - parallel fashion.

Answer:

• An image created by a reflection will always be congruent to its pre - image.
• An image and its pre - image are always the same distance from the line of reflection.
• If a point on the pre - image lies on the line of reflection, the image of that point is the same as the pre - image.
• The line of reflection is perpendicular to the line segments connecting corresponding vertices.