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Describe a sequence of transformations that moves the shaded pre-image onto the unshaded image to show that the figures are similar.

The shaded and unshaded triangles are similar, and we can demonstrate this by describing a sequence of transformations. Here’s the process to move the shaded triangle \( \triangle ABC \) onto the unshaded triangle \( \triangle A'B'C' \):

1. **Translation**: 
   Move the shaded triangle \( \triangle ABC \) such that point \( A \) aligns with point \( A' \). This can be done by translating all the vertices by the vector \( \overrightarrow{AA'} \). This translation ensures that \( A \) maps to \( A' \), while the relative positions of points \( B \) and \( C \) are preserved.

2. **Dilation**:
   After the translation, dilate the triangle with center \( A' \) and a scale factor. The scale factor can be found by comparing the lengths of corresponding sides in the triangles:
   \[
   \text{Scale factor} = \frac{\text{length of } A'B'}{\text{length of } AB} = \frac{\text{length of } A'C'}{\text{length of } AC}
   \]
   This dilation will enlarge the smaller triangle \( \triangle ABC \) to match the size of \( \triangle A'B'C' \), making them congruent.

Since these two transformations (translation and dilation) map the shaded triangle exactly onto the unshaded triangle, they are similar by definition. Similarity is established when figures can be mapped onto each other using a sequence of rigid transformations (like translation) and dilations.

Would you like more details on any step, or do you have any further questions?

Here are five related questions:
1. What properties define similar triangles?
2. How do you determine the scale factor for dilation in similar figures?
3. Why do translations not affect the size or shape of a figure?
4. What is the difference between congruence and similarity in geometry?
5. Can a triangle be mapped to another triangle using just a reflection? Why or why not?

**Tip:** In similarity transformations, angles remain unchanged, but the lengths of corresponding sides change by a constant ratio.

Learn how to demonstrate similarity between two triangles using translation and dilation. This problem covers key geometric concepts, such as similarity transformations, and provides step-by-step instructions for finding the correct scale factor. Suitable for students in Grades 8-10.

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