1. **State the problem:** We have triangle ABC with vertices $A(0,2)$, $B(-2,2)$, and $C(-2,0)$ in quadrant two and three. We need to create triangle DEF in quadrant one such that applying a translation, rotation, and reflection (in any order) maps DEF onto ABC. 2. **Understand transformations:** - **Translation:** moves a figure without rotating or flipping it. - **Rotation:** turns the figure around a point by a certain angle. - **Reflection:** flips the figure over a line (axis). 3. **Step 1: Choose triangle DEF in quadrant one.** Let's pick $D(1,1)$, $E(3,1)$, and $F(3,3)$ in quadrant one. 4. **Step 2: Find transformations to map DEF to ABC.** - First, reflect DEF over the line $y=x$ to swap coordinates: $$D'(1,1), E'(1,3), F'(3,3)$$ - Next, rotate $D'E'F'$ 90 degrees clockwise about the origin: $$D''(1,-1), E''(3,-1), F''(3,-3)$$ - Finally, translate $D''E''F''$ by vector $(-3,3)$: $$D'''(1-3,-1+3)=(-2,2), E'''(3-3,-1+3)=(0,2), F'''(3-3,-3+3)=(0,0)$$ 5. **Step 3: Check if $D'''E'''F'''$ matches $ABC$.** $A(0,2)$ matches $E'''(0,2)$ $B(-2,2)$ matches $D'''(-2,2)$ $C(-2,0)$ does not match $F'''(0,0)$, so adjust translation vector. 6. **Adjust translation vector to $(-3,2)$:** $$D'''(1-3,-1+2)=(-2,1), E'''(3-3,-1+2)=(0,1), F'''(3-3,-3+2)=(0,-1)$$ Still no match. 7. **Alternative approach:** Reflect DEF over $y=0$ (x-axis): $$D'(1,-1), E'(3,-1), F'(3,-3)$$ Rotate 90 degrees counterclockwise: $$D''(1,1), E''(1,3), F''(3,3)$$ Translate by $(-3,-1)$: $$D'''(-2,0), E'''(-2,2), F'''(0,2)$$ Now vertices match $C(-2,0)$, $B(-2,2)$, and $A(0,2)$. **Final answer:** Triangle DEF with vertices $D(1,1)$, $E(3,1)$, $F(3,3)$ in quadrant one can be mapped to triangle ABC by reflecting over the x-axis, rotating 90 degrees counterclockwise, then translating by $(-3,-1)$.