To show that two figures are similar, we must identify a sequence of transformations—translations, rotations, reflections, and dilations—that maps the smaller figure onto the larger one. For each pair of figures, the process involves describing the specific steps, including the type and order of transformations.

Let's break this down with a general method you can apply to any pair of figures:

Steps to demonstrate similarity through transformations:

1. Translation:

   • If the smaller figure is not already aligned with the larger one, start by translating it. A translation is a slide along a straight path that moves the figure without rotating it. Identify how far and in which direction the smaller figure needs to move to match a corresponding point in the larger figure.

2. Rotation:

   • If the orientation of the smaller figure differs from the larger one, apply a rotation. Rotations turn the figure around a fixed point, usually about the origin or a vertex. You need to determine the angle of rotation and whether it is clockwise or counterclockwise.

3. Reflection:

   • If the smaller figure is a mirror image of the larger one, use reflection. A reflection flips the figure over a line (axis of reflection). Identify the line of reflection needed, which could be the x-axis, y-axis, or any other axis.

4. Dilation:

   • Finally, use a dilation to scale the smaller figure up to the size of the larger one. A dilation enlarges (or reduces) the figure while keeping its shape the same. Determine the scale factor by comparing corresponding side lengths or distances between points.

Example: Showing similarity with a sequence of transformations

Consider two triangles where the smaller one is positioned at the origin.

1. Translation: Move the smaller triangle so that one of its vertices aligns with a corresponding vertex of the larger triangle.

2. Rotation: Rotate the smaller triangle about the aligned vertex until its orientation matches that of the larger triangle.

3. Dilation: Apply a dilation centered at the aligned vertex, with a scale factor determined by comparing corresponding side lengths, to make the smaller triangle the same size as the larger triangle.

Sequence Outline:

• Translate the smaller figure to a matching point on the larger figure.
• Rotate it so that corresponding angles are aligned.
• Reflect if needed to flip the figure.
• Dilate by a scale factor to adjust the size.

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Let me know if you'd like a specific example or need clarification!

Expansion Questions:

1. How do you determine the correct rotation angle for a pair of figures?
2. Can two figures be similar without using any reflection in the transformations?
3. How do you calculate the scale factor for a dilation when comparing two figures?
4. How do reflections affect the orientation of a figure in terms of similarity?
5. What is the role of corresponding angles and sides in determining similarity?

Tip:

Always verify that corresponding angles remain equal and side lengths are proportional after applying the transformations. This ensures the figures are indeed similar.