Question

which sequence of transformations could map △abc to △xyz?
○ a reflection across line m and a dilation
○ a dilation by $\frac{1}{4}$ and a reflection across line m
○ a rotation about c and a dilation
○ a dilation by $\frac{1}{4}$ and a translation

Explanation:

Step1: Analyze side lengths for dilation

First, find the scale factor. For side \( AC = 4 \) and \( XZ = 1 \), the scale factor \( k=\frac{XZ}{AC}=\frac{1}{4} \). Similarly, \( AB = 6 \), \( XY = 1.5 \), \( \frac{1.5}{6}=\frac{1}{4} \); \( BC = 5 \), \( YZ = 1.25 \), \( \frac{1.25}{5}=\frac{1}{4} \). So dilation by \( \frac{1}{4} \) is needed.

Step2: Analyze orientation and transformation

After dilation, the orientation of \( \triangle ABC \) (with \( AC \) horizontal) and \( \triangle XYZ \) (with \( XZ \) horizontal) – wait, no, actually, the position relative to line \( m \): a translation would move the dilated triangle to match \( \triangle XYZ \)'s position. Reflection or rotation? The angles match (marked angles), so after dilation by \( \frac{1}{4} \), a translation (not reflection/rotation) aligns them, as the orientation (angle positions) and side directions (after dilation) can be moved by translation. The other options: reflection would flip, but here the triangles have same "direction" after dilation. Rotation about \( C \) wouldn't align with \( XYZ \)'s position. So dilation by \( \frac{1}{4} \) and translation.

Answer:

a dilation by \(\frac{1}{4}\) and a translation