Question

which of the following sequences of transformations maps △uvw onto △abc?

sequence a
a 180° rotation about point v, then a translation along the directed line segment \\(\overline{vb}\\).

sequence b
a translation along the directed line segment \\(\overline{ua}\\), then a 180° rotation about point a.

Explanation:

To determine which sequence maps \( \triangle UVW \) onto \( \triangle ABC \), we analyze each sequence:

Analyzing Sequence A:

1. 180° rotation about \( V \): A 180° rotation about a point \( (x,y) \) transforms a point \( (a,b) \) to \( (2x - a, 2y - b) \). Rotating \( \triangle UVW \) 180° about \( V \) will flip the triangle’s orientation around \( V \).
2. Translation along \( \overrightarrow{VB} \): Translating the rotated triangle along the vector from \( V \) to \( B \) will move it to align with \( \triangle ABC \).

Analyzing Sequence B:

1. Translation along \( \overrightarrow{UA} \): Translating \( \triangle UVW \) along \( \overrightarrow{UA} \) moves \( U \) to \( A \), but a subsequent 180° rotation about \( A \) would not align the triangle with \( \triangle ABC \) (since the orientation and position after translation do not match the required final position).

By visualizing the grid and the triangles’ positions/orientations, Sequence A (180° rotation about \( V \), then translation along \( \overrightarrow{VB} \)) correctly maps \( \triangle UVW \) onto \( \triangle ABC \).

Answer:

Sequence A (A 180° rotation about point \( V \), then a translation along the directed line segment \( \overrightarrow{VB} \))