Question

one of the most important transformations is a 180° rotation about a point. these types of rotations are known as point reflections.
point reflections
a point reflection is a rotation of points in the plane about a given point by 180°. since it is exactly one - half of a full rotation, the direction of rotation does not matter.
exercise #1: using a straightedge and a compass only, find the image of \\( \overline{ab} \\) after a point reflection across or through point c. again, this is the same as a rotation about point c by 180°.
what must be true about the lengths \\( \overline{ab} \\) and \\( \overline{ab} \\)? why?

Explanation:

Step1: Recall the property of rotation

A rotation is a rigid transformation. Rigid transformations preserve the lengths of segments.

Step2: Apply the property to the given rotation

Since a \(180^{\circ}\) rotation (point reflection) is a rigid transformation, the length of the segment \(\overline{AB}\) and its image \(\overline{A'B'}\) after the rotation about point \(C\) will be equal.

Answer:

\(\overline{AB}\) and \(\overline{A'B'}\) have the same length. Because a \(180^{\circ}\) rotation (point reflection) is a rigid transformation, and rigid transformations preserve the lengths of segments.