1. **State the problem:** Reflect triangle ABC with vertices A(-5, 2), B(-6, -1), and C(-3, 0) over the line $x = -6$, then translate the image by the vector from $(0,0)$ to $(5,-1)$. 2. **Reflection over the line $x = -6$:** - The line $x = -6$ is vertical. - To reflect a point $(x,y)$ over $x = -6$, use the formula: $$x' = 2(-6) - x = -12 - x$$ $$y' = y$$ 3. **Apply reflection to each vertex:** - For $A(-5, 2)$: $$x'_A = -12 - (-5) = -12 + 5 = -7$$ $$y'_A = 2$$ So, $A' = (-7, 2)$ - For $B(-6, -1)$: $$x'_B = -12 - (-6) = -12 + 6 = -6$$ $$y'_B = -1$$ So, $B' = (-6, -1)$ - For $C(-3, 0)$: $$x'_C = -12 - (-3) = -12 + 3 = -9$$ $$y'_C = 0$$ So, $C' = (-9, 0)$ 4. **Translate the reflected triangle by vector $(5, -1)$:** - Translation formula: $$x'' = x' + 5$$ $$y'' = y' - 1$$ - For $A'(-7, 2)$: $$x''_A = -7 + 5 = -2$$ $$y''_A = 2 - 1 = 1$$ So, $A'' = (-2, 1)$ - For $B'(-6, -1)$: $$x''_B = -6 + 5 = -1$$ $$y''_B = -1 - 1 = -2$$ So, $B'' = (-1, -2)$ - For $C'(-9, 0)$: $$x''_C = -9 + 5 = -4$$ $$y''_C = 0 - 1 = -1$$ So, $C'' = (-4, -1)$ **Final coordinates of the vertices after reflection and translation:** $$A''(-2, 1), B''(-1, -2), C''(-4, -1)$$