1. **Stating the problem:** We have a mobile login screen layout composed of geometric shapes with given vertices: - Screen Container (rectangle): $(-1,1), (-1,8), (3,8), (3,1)$ - Login Button (triangle): $(1,2), (1.5,3), (2,2)$ - Username Field (rectangle): $(2,5), (2,6), (0,6), (0,5)$ We need to perform two transformations: - Translation: Move the entire layout 2 units right and 1 unit down. - Reflection: Reflect the translated layout about the line $y = -x$. 2. **Formulas and rules:** - Translation by vector $(a,b)$ moves each point $(x,y)$ to $(x+a, y+b)$. - Reflection about the line $y = -x$ transforms each point $(x,y)$ to $(-y, -x)$. 3. **Step 1: Translation by $(2, -1)$** (2 right, 1 down): - Screen Container vertices: $$(-1+2, 1-1) = (1,0), (-1+2, 8-1) = (1,7), (3+2, 8-1) = (5,7), (3+2, 1-1) = (5,0)$$ - Login Button vertices: $$(1+2, 2-1) = (3,1), (1.5+2, 3-1) = (3.5,2), (2+2, 2-1) = (4,1)$$ - Username Field vertices: $$(2+2, 5-1) = (4,4), (2+2, 6-1) = (4,5), (0+2, 6-1) = (2,5), (0+2, 5-1) = (2,4)$$ 4. **Step 2: Reflection about $y = -x$**: Apply $(x,y) \to (-y, -x)$ to each translated vertex. - Screen Container: $$(1,0) \to (0,-1), (1,7) \to (-7,-1), (5,7) \to (-7,-5), (5,0) \to (0,-5)$$ - Login Button: $$(3,1) \to (-1,-3), (3.5,2) \to (-2,-3.5), (4,1) \to (-1,-4)$$ - Username Field: $$(4,4) \to (-4,-4), (4,5) \to (-5,-4), (2,5) \to (-5,-2), (2,4) \to (-4,-2)$$ 5. **Final answer:** - Screen Container vertices after transformations: $(0,-1), (-7,-1), (-7,-5), (0,-5)$ - Login Button vertices after transformations: $(-1,-3), (-2,-3.5), (-1,-4)$ - Username Field vertices after transformations: $(-4,-4), (-5,-4), (-5,-2), (-4,-2)$ This completes the translation and reflection of the UI layout on the coordinate plane.