Question

geometry basics
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 distance relationship

For a point \(C\) and its image \(C'\) after reflection over a line (not shown in the general property description here, but based on reflection rules), the distance from \(C\) to the line of reflection (point \(P\) in the context of the problem, assuming \(P\) is on the line of reflection) and the distance from \(C'\) to the line of reflection (the same line) has the property that \(CP = C'P\).

Step3: Check each option

• Option 1: Point \(P\) is the mid - point of segment \(BB'\). In a reflection, the line of reflection is the perpendicular bisector of \(BB'\), not necessarily that \(P\) (a point on the line of reflection) is the mid - point of \(BB'\).
• Option 2: The sides of the pre - image and image are perpendicular to the line of reflection. In a reflection, the line of reflection is the perpendicular bisector of the segments joining pre - image and image points, not that the sides of the pre - image and image are perpendicular to the line of reflection.
• Option 3: The distance between \(C\) and \(P\) is equal to the distance between \(C'\) and \(P\). This is a fundamental property of reflection. If \(P\) is on the line of reflection, then for a point \(C\) and its image \(C'\) under reflection, the line of reflection (where \(P\) lies) is the perpendicular bisector of \(CC'\), so \(CP=C'P\).
• Option 4: The distance between \(C\) and \(C'\) is equal to twice the distance between \(C\) and \(P\). If \(CP = C'P\) (from reflection property \(CP=C'P\)), then \(CC'=CP + C'P=2CP\) is incorrect. The correct relationship is \(CP = C'P\) (not \(CC' = 2CP\)).

Answer:

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