1. **Problem Statement:** We are given two triangles, ABC and AED, with side lengths AB=8, BC=4, AC=6, and AE=4, ED=2, AD=3. We need to write two different sequences of transformations to show that these triangles are similar and define similarity. 2. **Definition of Similarity:** Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. 3. **Check Side Ratios:** Calculate ratios of corresponding sides: $$\frac{AB}{AE} = \frac{8}{4} = 2$$ $$\frac{BC}{ED} = \frac{4}{2} = 2$$ $$\frac{AC}{AD} = \frac{6}{3} = 2$$ All ratios are equal, so triangles are similar by the Side-Side-Side (SSS) similarity criterion. 4. **Sequence 1: Dilation and Translation** - Dilate triangle AED by a scale factor of 2 centered at point A. - This maps AE to AB, ED to BC, and AD to AC. - Then translate the dilated triangle so that point E coincides with point B. - This shows triangle AED maps onto triangle ABC, proving similarity. 5. **Sequence 2: Rotation and Dilation** - Rotate triangle AED around point A so that segment AE aligns with segment AB. - Dilate the rotated triangle by a scale factor of 2 centered at A. - This maps AED onto ABC, confirming similarity. Final answer: Triangles ABC and AED are similar by SSS similarity with scale factor 2.