Question

what are the transformations in the sequence that maps \\(\delta abc\\) onto \\(\delta def\\)?

1. dilate \\(\delta abc\\) by a scale factor of centered at \\(p\\).
2. translate along the vector
3. rotate about point to map \\(\overline{ba}\\) onto \\(\overline{ed}\\).
4. reflect over line

Explanation:

⚡ Using what you learned: dilations and scale factors · Rigid Transformations (Translation, Rotation, Reflection)

Step 1: Find the scale factor of the dilation

Compare the corresponding side lengths of the original triangle \(\triangle ABC\) and the intermediate dilated triangle \(\triangle A'B'C'\):

• Side \(AB = 5\) maps to side \(A'B' = 2.5\).
• Side \(BC = 9\) maps to side \(B'C' = 4.5\).

Calculate the scale factor \(k\):
\[k = \frac{A'B'}{AB} = \frac{2.5}{5} = 0.5\]

Step 2: Determine the translation vector

After dilating \(\triangle ABC\) centered at \(P\) to get \(\triangle A'B'C'\), we need to translate \(\triangle A'B'C'\) so that one of its vertices aligns with the target triangle \(\triangle DEF\).

Looking at the next step, we want to rotate about a point to map segment \(\overline{B''A''}\) onto \(\overline{ED}\). This indicates that vertex \(B''\) should land on vertex \(E\). Therefore, we translate \(\triangle A'B'C'\) along the vector from \(B'\) to \(E\):
\[\text{Vector } \vec{B'E}\]

This translation maps \(\triangle A'B'C'\) to \(\triangle A''B''C''\), where \(B''\) is at the same position as \(E\).

Step 3: Identify the center of rotation

Since vertex \(B''\) is already mapped onto vertex \(E\), we rotate the triangle about this shared point to align the adjacent sides.

• Center of rotation: \(E\) (or \(B''\))

This rotation maps segment \(\overline{B''A''}\) onto segment \(\overline{ED}\).

Step 4: Identify the line of reflection

After the rotation, segment \(\overline{B''A''}\) lies on segment \(\overline{ED}\). To complete the mapping of the entire triangle onto \(\triangle DEF\), we reflect the remaining part of the triangle over the line containing this aligned segment:

• Line of reflection: \(\overleftrightarrow{ED}\)

Answer:

1. Dilate \(\triangle ABC\) by a scale factor of \(0.5\) centered at \(P\).
2. Translate along the Vector \(\vec{B'E}\).
3. Rotate about Point \(E\) to map \(\overline{B''A''}\) onto \(\overline{ED}\).
4. Reflect over Line \(\overleftrightarrow{ED}\).