Question

quadrilateral ( abcd ) is the result of rotating quadrilateral ( abcd ) by ( 60^{circ} ) about the origin.
select all of the correct statements about the unchanged properties of quadrilateral ( abcd ) and quadrilateral ( abcd ).
choose all answers that apply:
a ( overline{bc} ) and ( overline{bc} ) are both parallel to the ( y )-axis.
b ( angle b ) and ( angle b ) have the same measures.
c ( d ) and ( d ) have the same coordinates.
d none of the above

Explanation:

Step1: Properties of Rotation

Rotation is a rigid transformation. Rigid transformations preserve side - lengths, angle - measures, parallelism, and collinearity.

Step2: Analyze Option A

Since rotation preserves parallelism, if \(BC\) is parallel to the \(y\) - axis (vertical line), then \(B'C'\) (the image of \(BC\) after rotation) is also parallel to the \(y\) - axis.

Step3: Analyze Option B

Since rotation is a rigid transformation, it preserves angle - measures. So, \(m\angle B=m\angle B'\).

Step4: Analyze Option C

A rotation about the origin changes the coordinates of a point (except when the point is the origin itself). Point \(D\) is not the origin, so \(D\) and \(D'\) do not have the same coordinates.

Answer:

A. \(\overline{BC}\) and \(\overline{B'C'}\) are both parallel to the \(y\) - axis; B. \(\angle B\) and \(\angle B'\) have the same measures.