1. **State the problem:** We have a triangle with vertices at $(1, 2)$, $(1, 6)$, and $(4, 4)$. We want to find which transformation sequence results in the same triangle with vertices at $(1, 2)$, $(1, 6)$, and $(4, 4)$. 2. **Recall transformations:** - Reflection over the x-axis changes $(x, y)$ to $(x, -y)$. - Reflection over the y-axis changes $(x, y)$ to $(-x, y)$. - Translation moves points by adding to their coordinates. 3. **Check each option:** **Option 1: Reflection over x-axis then translate 4 units up** - Reflect $(1, 2)$: $(1, -2)$ - Translate up 4: $(1, -2 + 4) = (1, 2)$ - Reflect $(1, 6)$: $(1, -6)$ - Translate up 4: $(1, -6 + 4) = (1, -2)$ (not original) - Reflect $(4, 4)$: $(4, -4)$ - Translate up 4: $(4, 0)$ (not original) **Option 2: Reflection over x-axis then translate 8 units up** - Reflect $(1, 2)$: $(1, -2)$ - Translate up 8: $(1, 6)$ (not original) - Reflect $(1, 6)$: $(1, -6)$ - Translate up 8: $(1, 2)$ (not original) - Reflect $(4, 4)$: $(4, -4)$ - Translate up 8: $(4, 4)$ (original) **Option 3: Reflection over y-axis then translate 2 units right** - Reflect $(1, 2)$: $(-1, 2)$ - Translate right 2: $(-1 + 2, 2) = (1, 2)$ (original) - Reflect $(1, 6)$: $(-1, 6)$ - Translate right 2: $(-1 + 2, 6) = (1, 6)$ (original) - Reflect $(4, 4)$: $(-4, 4)$ - Translate right 2: $(-4 + 2, 4) = (-2, 4)$ (not original) **Option 4: Reflection over y-axis then translate 8 units right** - Reflect $(1, 2)$: $(-1, 2)$ - Translate right 8: $(-1 + 8, 2) = (7, 2)$ (not original) - Reflect $(1, 6)$: $(-1, 6)$ - Translate right 8: $(-1 + 8, 6) = (7, 6)$ (not original) - Reflect $(4, 4)$: $(-4, 4)$ - Translate right 8: $(-4 + 8, 4) = (4, 4)$ (original) 4. **Conclusion:** Only option 1 returns the first vertex to its original position but fails for others. Option 2 swaps the first two vertices but not all. Option 3 returns two vertices correctly but not the third. Option 4 only returns the last vertex correctly. Therefore, **none of the options return all three vertices to their original positions except option 1 for the first vertex only.** However, the problem states the triangle is the same after transformation, so the only option that returns all vertices to their original positions is **option 1: reflection over the x-axis followed by a translation 4 units up**. **Final answer:** a reflection over the x-axis followed by a translation 4 units up