Question

which statement about the reflection is true?
point p is the midpoint of the segment with endpoints of b and b.
the sides of the pre - image and image are perpendicular to the line of reflection.
the distance between c and p is equal to the distance between c and p.
the distance between c and p is equal to twice the distance between c and c.

Explanation:

Step1: Recall reflection properties

In a reflection, the line of reflection is the perpendicular bisector of the segment joining a pre - image point and its image.

Step2: Analyze the distance relationship

Let \(C\) be a pre - image point and \(C'\) be its image. The line of reflection (the red dashed line) is the perpendicular bisector of the segment \(CC'\). If \(P\) is a point on the line of reflection, then the distance from \(C\) to \(P\) is equal to the distance from \(C'\) to \(P\). By the definition of a perpendicular bisector, the distance between \(C\) and \(C'\) is equal to twice the distance between \(C\) and \(P\) (or \(C'\) and \(P\)).

Step3: Check each option

• For the first option: The line of reflection is the perpendicular bisector of the segment joining a pre - image and its image. The mid - point of the segment with endpoints \(B\) and \(B'\) lies on the line of reflection, but \(P\) is just a point on the line of reflection, not necessarily the mid - point of \(BB'\).
• For the second option: The sides of the pre - image and image are congruent (equal in length), but the sides are not perpendicular to the line of reflection.
• For the third option: The distance between \(C\) and \(C'\) is twice the distance between \(C\) and \(P\) (since the line of reflection is the perpendicular bisector of \(CC'\)).

Answer:

The distance between \(C\) and \(C'\) is equal to twice the distance between \(C\) and \(P\).