Question

triangle \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) is the result of reflecting \\( \triangle a b c \\) across the line \\( l \\).

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:

\\( \square \\) \\( \overline { b c } \\) and \\( \overline { b ^ { prime } c ^ { prime } } \\) are both perpendicular to the \\( y \\)-axis.

\\( \square \\) \\( \angle b \\) and \\( \angle b ^ { prime } \\) have the same measures.

\\( \square \\) the perimeters of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) are the same.

none of the above

Explanation:

Step1: Properties of Reflection

Reflection is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures.

Step2: Analyze Option A

Since \(BC\) is horizontal (parallel to the \(x\) - axis), and after reflection \(B'C'\) is also horizontal (parallel to the \(x\) - axis). A line parallel to the \(x\) - axis is perpendicular to the \(y\) - axis.

Step3: Analyze Option B

Because reflection is a rigid transformation, \(\angle B\) and \(\angle B'\) (corresponding angles of the pre - image \(\triangle ABC\) and image \(\triangle A'B'C'\)) have the same measure.

Step4: Analyze Option C

Since reflection preserves side - lengths (\(AB = A'B'\), \(BC=B'C'\), \(AC = A'C'\)), the perimeter of \(\triangle ABC\) (\(P_{ABC}=AB + BC+AC\)) and the perimeter of \(\triangle A'B'C'\) (\(P_{A'B'C'}=A'B'+B'C'+A'C'\)) are the same.

Answer:

A. \(BC\) and \(B'C'\) are both perpendicular to the \(y\) - axis.
B. \(\angle B\) and \(\angle B'\) have the same measures.
C. The perimeters of \(\triangle ABC\) and \(\triangle A'B'C'\) are the same.