Question

3 practice 6 (from unit 1, lesson 16) a quadrilateral has rotation symmetry that can take any of its vertices to any of its other vertices. select all conclusions that we can reach from this. a all sides of the quadrilateral have the same length. b all angles of the quadrilateral have the same measure. c all rotations take one half of the quadrilateral to the other half of the quadrilateral.

Explanation:

Step1: Analyze rotation - symmetry property

If a quadrilateral has rotation symmetry that can take any of its vertices to any of its other vertices, then the distance from the center of rotation to each vertex is the same. This implies that the lengths of the sides connecting the vertices are equal. Because when we rotate one vertex to another, the side - length between them must be invariant under this rotation. So, all sides of the quadrilateral have the same length.

Step2: Consider angle - measure

Just because a quadrilateral has vertex - to - vertex rotation symmetry does not necessarily mean that all angles have the same measure. For example, a rhombus has rotation symmetry (a 180 - degree rotation about its center takes vertices to vertices), but its angles are not all equal (except in the special case of a square).

Step3: Analyze rotation and halves

Not all rotations take one half of the quadrilateral to the other half. A general rotation that maps vertices to vertices does not imply a reflection - like property that divides the quadrilateral into two equal halves.

Answer:

A. All sides of the quadrilateral have the same length.