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: Calculate the scale factor

The length of \(AC = 4\), and the length of \(XZ=1.25\). The scale factor \(k=\frac{1.25}{5}=\frac{1}{4}\). A dilation is needed to change the size of \(\triangle ABC\) to \(\triangle XYZ\).

Step2: Analyze the orientation

By observing the angles and the position of the triangles relative to line \(m\), a reflection across line \(m\) is required to get the correct orientation of \(\triangle XYZ\) from \(\triangle ABC\) after dilation. A rotation about \(C\) would not give the correct orientation as per the figure. A translation would not change the orientation in the way shown (a reflection is more appropriate for the orientation change relative to line \(m\)). If we first dilate by \(\frac{1}{4}\), the side lengths would match (\(6\times\frac{1}{4} = 1.5\), \(4\times\frac{1}{4}=1\) (but \(XZ = 1.25\) which is \(5\times\frac{1}{4}\)), and then reflect across line \(m\) (the reflection changes the orientation).

Answer:

a reflection across line \(m\) and a dilation