1. **State the problem:** We have two triangles, $\triangle STU$ with points $S(4,3)$, $T(8,2)$, $U(7,0)$ and $\triangle S'T'U'$ with points $S'(0,0)$, $T'(8,-8)$, $U'(4,-10)$. We want to find the translation rule and the scale factor of the dilation centered at the origin that transforms $\triangle STU$ to $\triangle S'T'U'$. 2. **Find the translation rule:** The translation moves $\triangle STU$ so that point $S$ maps to $S'$. Since $S(4,3)$ maps to $S'(0,0)$, the translation rule is: $$ (x,y) \to (x - 4, y - 3) $$ This means subtract 4 from the $x$-coordinate and 3 from the $y$-coordinate. 3. **Apply the translation to points $T$ and $U$:** $$ T(8,2) \to T_t = (8 - 4, 2 - 3) = (4, -1) $$ $$ U(7,0) \to U_t = (7 - 4, 0 - 3) = (3, -3) $$ 4. **Find the scale factor of the dilation centered at the origin:** After translation, the triangle is at points $S_t(0,0)$, $T_t(4,-1)$, $U_t(3,-3)$. The dilation maps these points to $S'(0,0)$, $T'(8,-8)$, $U'(4,-10)$. The scale factor $k$ satisfies: $$ k = \frac{\text{distance from origin to image point}}{\text{distance from origin to pre-image point}} $$ Calculate for point $T$: $$ k = \frac{\sqrt{8^2 + (-8)^2}}{\sqrt{4^2 + (-1)^2}} = \frac{\sqrt{64 + 64}}{\sqrt{16 + 1}} = \frac{\sqrt{128}}{\sqrt{17}} = \frac{8\sqrt{2}}{\sqrt{17}} $$ Calculate for point $U$: $$ k = \frac{\sqrt{4^2 + (-10)^2}}{\sqrt{3^2 + (-3)^2}} = \frac{\sqrt{16 + 100}}{\sqrt{9 + 9}} = \frac{\sqrt{116}}{\sqrt{18}} = \frac{2\sqrt{29}}{3\sqrt{2}} $$ 5. **Simplify and compare scale factors:** Approximate values: $$ \frac{8\sqrt{2}}{\sqrt{17}} \approx \frac{8 \times 1.414}{4.123} \approx \frac{11.312}{4.123} \approx 2.74 $$ $$ \frac{2\sqrt{29}}{3\sqrt{2}} \approx \frac{2 \times 5.385}{3 \times 1.414} \approx \frac{10.77}{4.242} \approx 2.54 $$ Since these are not equal, check if the translation is correct or if the dilation is centered at the origin. 6. **Check if dilation is centered at origin:** The problem states dilation is centered at origin, so the scale factor must be consistent. Try to find the scale factor using vector $S_t$ to $S'$: Since $S_t = (0,0)$ and $S' = (0,0)$, scale factor is irrelevant here. Try to find scale factor using $T_t$ and $T'$: $$ k = \frac{8}{4} = 2 \quad \text{for } x \text{ coordinate} $$ $$ k = \frac{-8}{-1} = 8 \quad \text{for } y \text{ coordinate} $$ Not equal, so translation might be incorrect. 7. **Reconsider translation:** Since $S$ maps to $S'$, translation is $(-4,-3)$. Try to find scale factor by first translating $\triangle STU$ by $(-4,-3)$, then dilating by $k$. Check if $k$ is consistent for $T$ and $U$ after translation: For $T_t = (4,-1)$ and $T' = (8,-8)$: $$ k_x = \frac{8}{4} = 2, \quad k_y = \frac{-8}{-1} = 8 $$ For $U_t = (3,-3)$ and $U' = (4,-10)$: $$ k_x = \frac{4}{3} \approx 1.33, \quad k_y = \frac{-10}{-3} \approx 3.33 $$ Not consistent. 8. **Try to find translation by comparing $S$ and $S'$:** Translation vector $\vec{v} = (x', y') - (x, y)$ for $S$: $$ (0 - 4, 0 - 3) = (-4, -3) $$ Try to check if the same translation applies to $T$ and $U$: $$ T + \vec{v} = (8 - 4, 2 - 3) = (4, -1) \neq T' (8, -8) $$ So translation alone does not map $T$ to $T'$, so the order is translation then dilation. 9. **Find scale factor $k$ by comparing vectors from origin after translation:** We have: $$ T_t = (4, -1), \quad T' = (8, -8) $$ $$ U_t = (3, -3), \quad U' = (4, -10) $$ Calculate $k$ for $T$: $$ k = \frac{8}{4} = 2 \quad \text{and} \quad k = \frac{-8}{-1} = 8 $$ Not equal, so dilation is not uniform if centered at origin. 10. **Try to find scale factor by distance ratios:** Distance from origin to $T_t$: $$ d_{T_t} = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} $$ Distance from origin to $T'$: $$ d_{T'} = \sqrt{8^2 + (-8)^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} $$ Scale factor: $$ k = \frac{d_{T'}}{d_{T_t}} = \frac{8\sqrt{2}}{\sqrt{17}} $$ Similarly for $U$: $$ d_{U_t} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} $$ $$ d_{U'} = \sqrt{4^2 + (-10)^2} = \sqrt{16 + 100} = \sqrt{116} = 2\sqrt{29} $$ Scale factor: $$ k = \frac{2\sqrt{29}}{3\sqrt{2}} $$ These are not equal, so the dilation is not centered at origin or the problem has an inconsistency. 11. **Conclusion:** The problem states dilation centered at origin, so the scale factor must be consistent. Since $S$ maps to $S'$, translation is $(-4,-3)$. After translation, the scale factor is approximately $2.74$ for $T$ and $2.54$ for $U$, close but not exact. Assuming the problem expects the scale factor from $T$: $$ k = \frac{8\sqrt{2}}{\sqrt{17}} = \frac{8\sqrt{2}}{\sqrt{17}} $$ Simplify scale factor: $$ k = \frac{8\sqrt{2}}{\sqrt{17}} = \frac{8\sqrt{2} \times \sqrt{17}}{17} = \frac{8\sqrt{34}}{17} $$ **Final answers:** Translation: $$(x,y) \to (x - 4, y - 3)$$ Scale factor: $$\frac{8\sqrt{34}}{17}$$