1. **State the problem:** (a) Translate triangle T by the vector $\begin{pmatrix}0 \\ -2\end{pmatrix}$. (b) Describe the single transformation that maps triangle T onto triangle U. (c) Solve the equations: (i) $8x + 7 = 39$ (ii) $2(5y - 1) = 24$ 2. **Translation of triangle T by vector $(0, -2)$:** Translation means moving every point of the triangle down by 2 units (since the vector is $0$ in $x$ and $-2$ in $y$). 3. **Coordinates of triangle T vertices:** Given approximately as $(-3,1)$, $(-2,1)$, and $(-3,2)$. 4. **Apply translation:** Each vertex $(x,y)$ becomes $(x, y - 2)$. - $(-3,1) \to (-3, 1 - 2) = (-3, -1)$ - $(-2,1) \to (-2, 1 - 2) = (-2, -1)$ - $(-3,2) \to (-3, 2 - 2) = (-3, 0)$ 5. **Compare with triangle U vertices:** Triangle U vertices are approximately $(2,-1)$, $(3,-2)$, and $(1,-2)$. 6. **Describe the single transformation mapping T onto U:** The translation alone does not map T onto U because the translated T vertices are $(-3,-1)$, $(-2,-1)$, $(-3,0)$, which do not match U. Observing the positions, triangle U is a reflection and translation of T. 7. **Check reflection:** Reflect triangle T across the $y$-axis: - $(-3,1) \to (3,1)$ - $(-2,1) \to (2,1)$ - $(-3,2) \to (3,2)$ Then translate down by 3 units: - $(3,1) \to (3, -2)$ - $(2,1) \to (2, -2)$ - $(3,2) \to (3, -1)$ These points are close to U's vertices $(2,-1)$, $(3,-2)$, $(1,-2)$ but not exact. Alternatively, the transformation is a rotation of 180 degrees about the origin: - $(-3,1) \to (3,-1)$ - $(-2,1) \to (2,-1)$ - $(-3,2) \to (3,-2)$ Then translate left by 1 unit: - $(3,-1) \to (2,-1)$ - $(2,-1) \to (1,-1)$ - $(3,-2) \to (2,-2)$ This matches U's vertices approximately. Hence, the single transformation is a rotation of 180 degrees about the origin followed by a translation of $(-1,0)$. 8. **Solve (a) $8x + 7 = 39$:** Subtract 7 from both sides: $$8x + 7 - 7 = 39 - 7$$ $$8x = 32$$ Divide both sides by 8: $$\cancel{8}x = \frac{32}{\cancel{8}}$$ $$x = 4$$ 9. **Solve (b) $2(5y - 1) = 24$:** Divide both sides by 2: $$\cancel{2}(5y - 1) = \frac{24}{\cancel{2}}$$ $$5y - 1 = 12$$ Add 1 to both sides: $$5y - 1 + 1 = 12 + 1$$ $$5y = 13$$ Divide both sides by 5: $$\cancel{5}y = \frac{13}{\cancel{5}}$$ $$y = \frac{13}{5} = 2.6$$ **Final answers:** (a) Translated triangle T vertices: $(-3,-1)$, $(-2,-1)$, $(-3,0)$. (b) Single transformation mapping T onto U: rotation of 180 degrees about the origin followed by translation by $(-1,0)$. (c) $x = 4$ (d) $y = \frac{13}{5}$ or $2.6$