Triangles A B C and X Y Z is congruent. Triangle X Y Z is slightly higher and to the right of triangle A B C. Triangle A B C is reflected across a line to form triangle X Y Z.

Two rigid transformations are used to map ΔABC to ΔXYZ. The first is a translation of vertex A to vertex X. What is the second transformation?

a reflection across the line containing AB
a reflection across the line containing AC
a rotation about point A
a rotation about point B

1 answer

Since triangle \( ABC \) is reflected across a line to form triangle \( XYZ \), and the first transformation is a translation of vertex \( A \) to vertex \( X \), we can determine the second transformation by considering how the specific points correspond after the translation.

When vertex \( A \) translates to vertex \( X \), this creates a new orientation for the triangle, but the second transformation must also ensure that the other corresponding vertices (\( B \) and \( C \)) map correctly to \( Y \) and \( Z \), respectively.

Since the problem states that triangle \( XYZ \) is formed by a reflection, and given that \( A \) translated to \( X \) implies that a flip or mirroring of the orientation is necessary, the second transformation must be a reflection across a line that ensures the remaining vertices match correctly.

To determine the correct reflection line, we look at the possible answers:

  1. Reflection across the line containing AB: This would mirror across the line extending from points \( A \) to \( B \), which wouldn’t correctly map \( B \) and \( C \) to \( Y \) and \( Z \) respectively.

  2. Reflection across the line containing AC: This would mirror across the line extending from points \( A \) to \( C \). This could potentially map \( B \) to \( Y \), depending on angles, and \( C \) to \( Z \).

  3. Rotation about point A: This does not yield the required mapping when considering the need for congruence and position.

  4. Rotation about point B: Similar to the above, this would not work as a second transformation in this context.

Thus, the correct second transformation, to maintain the congruence of triangles and the corresponding positions, would be:

a reflection across the line containing AC.