Question

1. which of the following is not a property of all rigid motions? (1) the length of a line segment and its image will be the same. (2) a line will be mapped to an image line that is parallel to the pre - image. (3) an angle will be mapped to an image angle with the same measure. (4) two perpendicular lines will have images that are also perpendicular. 4. point c is reflected across \\( \overline{ab} \\) such that it maps onto point d. points c and d are then connected to form \\( \overline{cd} \\). which of the following must be true? (1) \\( \overline{cd} \\) bisects \\( \overline{ab} \\) (2) \\( \overline{cd} \\) is perpendicular to \\( \overline{ab} \\) (3) \\( ab = cd \\) (4) \\( \overline{cd} \\) is parallel to \\( \overline{ab} \\)

Explanation:

3.

Step1: Recall properties of rigid motions

Rigid motions (translations, rotations, reflections) preserve lengths (so (1) is a property), preserve angle measures (so (3) is a property), and preserve perpendicularity (if two lines are perpendicular, their images under rigid - motion are perpendicular, so (4) is a property).
For a reflection, a line and its image are not parallel (they are symmetric with respect to the line of reflection). For a rotation, a line and its image are not parallel (except for a \(360^{\circ}\) rotation which is the identity).

4.

Step1: Use the property of reflection

When a point \(C\) is reflected across a line \(\overline{AB}\) to get a point \(D\), the line segment \(\overline{CD}\) is perpendicular to the line of reflection \(\overline{AB}\).
The line of reflection \(\overline{AB}\) is the perpendicular bisector of \(\overline{CD}\) (by the definition of reflection: the line of reflection is the set of points equidistant from \(C\) and \(D\)). There is no guarantee that \(\overline{CD}\) bisects \(\overline{AB}\) (unless \(\overline{AB}\) is also symmetric with respect to \(\overline{CD}\)), \(AB = CD\) (lengths of \(\overline{AB}\) and \(\overline{CD}\) are not necessarily related in that way), or \(\overline{CD}\parallel\overline{AB}\) (since the line of reflection is the perpendicular bisector of \(\overline{CD}\)).

Answer:

1. (2)
2. (2)