Question

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

1. dilate \\(\triangle abc\\) by a scale factor of 1/2 centered at p.
2. translate along the vector...
3. rotate about point...
4. reflect over line...

Explanation:

⚡ Using what you learned: Congruence and Similarity via Transformations

Step 1: Determine the scale factor of dilation

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

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

Calculate the scale factor \(k\):

$$k = \frac{A'B'}{AB} = \frac{2.5}{5} = \frac{1}{2}$$

Thus, the first step is to dilate \(\triangle ABC\) by a scale factor of \(\frac{1}{2}\) centered at \(P\).

Step 2: Determine the translation vector

After dilation, the intermediate triangle \(\triangle A'B'C'\) must be translated so that one of its vertices coincides with the corresponding vertex of the target triangle \(\triangle DEF\).

Looking at the orientation and side lengths of \(\triangle DEF\):

• Side \(ED = 2.5\) corresponds to \(A'B' = 2.5\)
• Side \(EF = 4.5\) corresponds to \(B'C' = 4.5\)

This means vertex \(B'\) corresponds to vertex \(E\), and vertex \(A'\) corresponds to vertex \(D\).

To map the intermediate triangle onto the target position, we translate along the vector that maps \(B'\) to \(E\):

$$\vec{B'E}$$

Step 3: Determine the rotation

After translating \(B'\) to \(E\), the segment \(B''A''\) (which is now starting at \(E\)) must be rotated to align with the corresponding segment \(ED\) of \(\triangle DEF\).

Therefore, we rotate about the common point:

$$\text{Point } E$$

This aligns segment \(B''A''\) onto segment \(ED\).

Step 4: Determine the reflection line

Once \(B''A''\) is aligned with \(ED\), the remaining vertex \(C''\) lies on the opposite side of the line containing \(ED\) compared to vertex \(F\).

To complete the mapping onto \(\triangle DEF\), reflect the triangle over the line containing the aligned segment:

$$\text{Line } ED$$

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Answer:

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