Question

1. three line segments form the letter z. rotate the letter z counterclockwise around the midpoint of segment bc by 180 degrees. describe the result.

type a response

Explanation:

Step1: Understand rotation property

A \(180^{\circ}\) rotation about a point is equivalent to a point - reflection (central inversion). For any point \(P\) and a center of rotation \(O\), if we rotate \(P\) \(180^{\circ}\) about \(O\), the image \(P'\) is such that \(O\) is the mid - point of the segment \(PP'\).

Step2: Analyze the rotation of the letter \(Z\)

Let \(M\) be the mid - point of \(BC\).

• When we rotate segment \(AB\) \(180^{\circ}\) about \(M\), the image of segment \(AB\) will be parallel to \(CD\) (since a \(180^{\circ}\) rotation preserves parallelism for non - collinear segments).
• When we rotate segment \(CD\) \(180^{\circ}\) about \(M\), the image of segment \(CD\) will be parallel to \(AB\).
• The segment \(BC\) is rotated onto itself because \(M\) (the mid - point of \(BC\)) is the center of rotation. A \(180^{\circ}\) rotation of a segment about its mid - point maps the segment onto itself.

Answer:

The result of rotating the letter \(Z\) counter - clockwise \(180^{\circ}\) around the mid - point of segment \(BC\) is the same letter \(Z\) (the letter \(Z\) is invariant under a \(180^{\circ}\) rotation about the mid - point of segment \(BC\)).