Question

triangle \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) is the result of rotating \\( \triangle a b c \\) by \\( - 135 ^ { circ } \\) about point a.
select all of the correct statements about the unchanged properties of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\).
choose all answers that apply:
a \\( a \\) and \\( a ^ { prime } \\) have the same coordinates.
b the perimeters of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) are the same.
c \\( \angle c \\) and \\( \angle c ^ { prime } \\) have the same measures.
d none of the above

Explanation:

Step1: Properties of rotation

Rotation is a rigid transformation. In a rotation about a point, the center of rotation (point \(A\) in this case) does not move. So, the coordinates of \(A\) and \(A'\) (since \(A\) is the center of rotation) are the same.

Step2: Effect on perimeter

Rigid - transformations (rotations, translations, reflections) preserve the lengths of the sides of a figure. The perimeter of a triangle is the sum of the lengths of its sides. If \(AB = A'B'\), \(BC=B'C'\), and \(AC = A'C'\) (because of the rotation, which is a rigid transformation), then \(P_{\triangle ABC}=AB + BC+AC\) and \(P_{\triangle A'B'C'}=A'B'+B'C'+A'C'\). So, the perimeters of \(\triangle ABC\) and \(\triangle A'B'C'\) are the same.

Step3: Effect on angles

Rigid - transformations also preserve the measures of angles. For any two corresponding angles (such as \(\angle C\) and \(\angle C'\)) in the pre - image (\(\triangle ABC\)) and the image (\(\triangle A'B'C'\)) of a rotation (a rigid transformation), \(m\angle C=m\angle C'\).

Answer:

A. \(A\) and \(A'\) have the same coordinates, B. The perimeters of \(\triangle ABC\) and \(\triangle A'B'C'\) are the same, C. \(\angle C\) and \(\angle C'\) have the same measures.