Question

which rigid transformation would map \\( \triangle abc \\) to \\( \triangle abf \\)?
\\( \bigcirc \\) a rotation about point a
\\( \bigcirc \\) a reflection across the line containing \\( \overline{cb} \\)
\\( \bigcirc \\) a reflection across the line containing \\( \overline{ba} \\)
\\( \bigcirc \\) a rotation about point b

Explanation:

Step1: Analyze rotation about point A

If we rotate about point A, the position of \( \triangle ABC\) will not map to \( \triangle ABF\) as the orientation and position relative to point A won't match the required transformation.

Step2: Analyze reflection across line \( \overline{CB}\)

Reflecting across the line containing \( \overline{CB}\) will not map \( \triangle ABC\) to \( \triangle ABF\) because the symmetry axis \( \overline{CB}\) does not align the two triangles.

Step3: Analyze reflection across line \( \overline{BA}\)

A reflection across the line containing \( \overline{BA}\) will map \( \triangle ABC\) to \( \triangle ABF\). Because the line \( \overline{BA}\) acts as the perpendicular - bisector of the segment joining \(C\) and \(F\), and \(AB\) is common to both triangles.

Step4: Analyze rotation about point B

Rotating about point B, the position of \( \triangle ABC\) will not map to \( \triangle ABF\) as the orientation and position relative to point B won't match the required transformation.

Answer:

a reflection across the line containing \( \overline{BA}\)