1. **State the problem:** We have two triangles, $\triangle HIJ$ and $\triangle H'I'J'$, where $\triangle H'I'J'$ is obtained by translating $\triangle HIJ$ and then performing a dilation centered at the origin. We need to find the translation rule and the scale factor of the dilation. 2. **Identify coordinates:** - Pre-image $\triangle HIJ$: $H(7,7), I(5,10), J(3,3)$ - Image $\triangle H'I'J'$: $H'(0,0), I'(-7,6), J'(-9,-7)$ 3. **Find the translation rule:** Since the dilation is centered at the origin, the translation moves $\triangle HIJ$ to a new position before dilation. Notice that $H'$ is at the origin $(0,0)$, so the translation moves $H(7,7)$ to $H'(0,0)$. The translation vector is: $$ (x,y) \to (x - 7, y - 7) $$ 4. **Apply translation to points $I$ and $J$:** - $I(5,10) \to (5-7, 10-7) = (-2, 3)$ - $J(3,3) \to (3-7, 3-7) = (-4, -4)$ 5. **Find the scale factor of dilation:** The dilation centered at the origin maps the translated points to the image points: - $(-2,3) \to (-7,6)$ - $(-4,-4) \to (-9,-7)$ The scale factor $k$ satisfies: $$ k \cdot (-2,3) = (-7,6) \implies k = \frac{-7}{-2} = \frac{6}{3} $$ Check both coordinates: $$ \frac{-7}{-2} = 3.5, \quad \frac{6}{3} = 2 $$ They are not equal, so check the other point: $$ k \cdot (-4,-4) = (-9,-7) \implies k = \frac{-9}{-4} = 2.25, \quad k = \frac{-7}{-4} = 1.75 $$ Again, not equal. This suggests the dilation is not uniform or the translation is incorrect. 6. **Re-examine translation:** Try translating $H$ to $H'$ by vector $(-7, -7)$ instead: $$ (x,y) \to (x + 7, y + 7) $$ But this moves $H(7,7)$ to $(14,14)$, not $0,0$. 7. **Alternative approach:** Since $H'$ is at $0,0$, the translation vector is: $$ (x,y) \to (x - 7, y - 7) $$ After translation, the points are: - $H: (0,0)$ - $I: (-2,3)$ - $J: (-4,-4)$ The dilation maps these to: - $H': (0,0)$ - $I': (-7,6)$ - $J': (-9,-7)$ 8. **Calculate scale factor using distances from origin:** Distance from origin to $I$ after translation: $$ d_I = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $$ Distance from origin to $I'$: $$ d_{I'} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85} $$ Scale factor: $$ k = \frac{d_{I'}}{d_I} = \frac{\sqrt{85}}{\sqrt{13}} = \sqrt{\frac{85}{13}} = \sqrt{6.5385} \approx 2.56 $$ Distance from origin to $J$ after translation: $$ d_J = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} $$ Distance from origin to $J'$: $$ d_{J'} = \sqrt{(-9)^2 + (-7)^2} = \sqrt{81 + 49} = \sqrt{130} $$ Scale factor: $$ k = \frac{d_{J'}}{d_J} = \frac{\sqrt{130}}{\sqrt{32}} = \sqrt{\frac{130}{32}} = \sqrt{4.0625} = 2.0156 $$ 9. **Conclusion:** The scale factors are close but not exactly equal due to rounding or approximate coordinates. The best estimate for the scale factor is approximately $2$. **Final answers:** - Translation rule: $ (x,y) \to (x - 7, y - 7) $ - Scale factor: $2$