1. **State the problem:** We have triangle ABC with points A(0,4), B(0,0), C(4,0) and its image A'B'C' with points A'(5,7), B'(5,3), C'(9,3) after a translation. We need to find the translation rule $T_{a,b}$ that maps ABC to A'B'C'. 2. **Recall the translation rule:** A translation $T_{a,b}$ moves every point $(x,y)$ to $(x+a, y+b)$. 3. **Find the translation vector:** - For point A: from $(0,4)$ to $(5,7)$, the horizontal shift is $5 - 0 = 5$ and vertical shift is $7 - 4 = 3$. - For point B: from $(0,0)$ to $(5,3)$, horizontal shift $5 - 0 = 5$, vertical shift $3 - 0 = 3$. - For point C: from $(4,0)$ to $(9,3)$, horizontal shift $9 - 4 = 5$, vertical shift $3 - 0 = 3$. 4. **Confirm all points have the same translation vector:** All points shift by $(5,3)$. 5. **Write the translation rule:** $$T_{5,3}(x,y) = (x+5, y+3)$$ 6. **Answer:** The correct translation is option B: $T_{5,3}(\triangle ABC)$.