1. **Problem 1: Find a sequence of rigid motions and dilations that takes square ABCD to square EFGH.** - Given: Square ABCD with side length 5. - Target: Square EFGH with side length 2, rotated 45 degrees. 2. **Step 1: Understand the problem.** - We want to transform ABCD into EFGH using rigid motions (translations, rotations, reflections) and dilations (scaling). 3. **Step 2: Use dilation to change size.** - The scale factor for dilation is the ratio of side lengths: $$\text{scale factor} = \frac{2}{5} = 0.4$$ 4. **Step 3: Apply dilation centered at a convenient point (e.g., center of ABCD).** - This reduces the square ABCD to a smaller square with side length 2. 5. **Step 4: Apply rotation to match orientation.** - Rotate the smaller square by 45 degrees to match the diamond shape of EFGH. 6. **Step 5: Apply translation if needed to align positions.** - Translate the rotated square to the position of EFGH. --- 7. **Problem 2: Quadrilaterals Q and P are similar.** - Given: Triangle P with sides 4, 3, 2. - Triangle Q with sides 5 and 2.5 corresponding to sides 4 and 2 in P. 8. **Step 1: Find scale factor from P to Q.** - Use corresponding sides 4 (P) and 5 (Q): $$\text{scale factor} = \frac{5}{4} = 1.25$$ - Check with sides 2 (P) and 2.5 (Q): $$\frac{2.5}{2} = 1.25$$ - So scale factor from P to Q is 1.25. 9. **Step 2: Find scale factor from Q to P.** - This is the reciprocal: $$\text{scale factor} = \frac{1}{1.25} = 0.8$$ --- **Final answers:** 1. Sequence: Dilate ABCD by 0.4 centered at its center, rotate by 45 degrees, then translate to EFGH. 2a. Scale factor from P to Q is 1.25. 2b. Scale factor from Q to P is 0.8.