1. **Problem statement:** We have two triangles, A and B, on a coordinate grid. Triangle A is translated to form triangle B. We need to find the values of $d$, $e$, and $f$ which are coordinates of vertices of these triangles. 2. **Understanding translation:** Translation moves every point of a shape by the same amount in the x and y directions. If a point $(x, y)$ is translated by $(h, k)$, the new point is $(x+h, y+k)$. 3. **Given points:** Triangle A vertices: $(-11, -5)$, $(-7, 2)$, and $(f, -3)$. Triangle B vertices: $(2, d)$, $(6, 12)$, and $(9, e)$. 4. **Find translation vector:** Compare corresponding points: - From $(-11, -5)$ to $(2, d)$: translation vector is $(2 - (-11), d - (-5)) = (13, d + 5)$. - From $(-7, 2)$ to $(6, 12)$: translation vector is $(6 - (-7), 12 - 2) = (13, 10)$. Since translation is the same for all points, $d + 5 = 10$ so $d = 5$. 5. **Find $f$ and $e$:** Using the translation vector $(13, 10)$: - For point $(f, -3)$ in A, the corresponding point in B is $(9, e)$. - So, $9 = f + 13$ which gives $f = 9 - 13 = -4$. - Also, $e = -3 + 10 = 7$. 6. **Final answers:** $$d = 5, \quad e = 7, \quad f = -4$$ Note: The user provided $d=8$, $e=10$, $f=-4$ but based on translation consistency, $d=5$, $e=7$, $f=-4$ is correct.