1. **State the problem:** We have two triangles, $\triangle TUV$ with vertices $T(4,4), U(2,2), V(6,2)$ and $\triangle T'U'V'$ with vertices $T'(0,0), U'(-10,-10), V'(10,-10)$. We want to find the translation rule and the scale factor of the dilation centered at the origin that maps $\triangle TUV$ to $\triangle T'U'V'$. 2. **Find the translation rule:** Translation moves every point by the same vector. Since $T$ moves to $T'$, the translation vector is: $$ (x,y) \to (x - 4, y - 4) $$ because $4 - 4 = 0$ and $4 - 4 = 0$ to map $T(4,4)$ to $T'(0,0)$. 3. **Apply translation to other points:** - $U(2,2)$ translates to $(2 - 4, 2 - 4) = (-2, -2)$ - $V(6,2)$ translates to $(6 - 4, 2 - 4) = (2, -2)$ 4. **Find the scale factor of dilation centered at origin:** After translation, the points $(-2,-2)$ and $(2,-2)$ map to $U'(-10,-10)$ and $V'(10,-10)$ respectively by dilation. The scale factor $k$ satisfies: $$ k = \frac{\text{distance from origin to } U'}{\text{distance from origin to translated } U} = \frac{\sqrt{(-10)^2 + (-10)^2}}{\sqrt{(-2)^2 + (-2)^2}} = \frac{\sqrt{200}}{\sqrt{8}} = \frac{10\sqrt{2}}{2\sqrt{2}} = 5 $$ 5. **Summary:** - Translation rule: $ (x,y) \to (x - 4, y - 4) $ - Scale factor of dilation centered at origin: $5$ Thus, the transformation is a translation by $(-4,-4)$ followed by a dilation centered at the origin with scale factor $5$.