Question

triangle ( \triangle abc ) is the result of reflecting ( \triangle abc ) across the line ( l ).
select all of the correct statements about the unchanged properties of ( \triangle abc ) and ( \triangle abc ).
choose all answers that apply
a ( overline{bc} ) and ( overline{bc} ) 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 abc ) are the same.
d none of the above

Explanation:

Step1: Analyze property of reflection

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

Step2: Check each option

• Option A:

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

• Option B:

Because of the reflection (a rigid transformation), \(\angle B\) and \(\angle B'\) are congruent (have the same measure).

• Option C:

Since reflection is a rigid transformation, \(AB = A'B'\), \(BC=B'C'\), and \(AC = A'C'\). The perimeter of \(\triangle ABC=AB + BC+AC\) and the perimeter of \(\triangle A'B'C'=A'B'+B'C'+A'C'\). So, the perimeters of \(\triangle ABC\) and \(\triangle A'B'C'\) are the same.

Answer:

A. \(\overline{BC}\) and \(\overline{B'C'}\) 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.