1. **Problem:** HIJK undergoes a dilation with a scale factor of 3 centered at the origin. Determine which statements are true. 2. **Understanding dilation:** A dilation centered at the origin with scale factor $k$ transforms any point $(x,y)$ to $(kx, ky)$. Important rules: - Lengths are multiplied by $k$. - Angle measures remain unchanged. 3. **Analyze each statement:** - a. $3(m\angle K) = m\angle K'$: Angles do not change under dilation, so $m\angle K = m\angle K'$. Multiplying $m\angle K$ by 3 is false. - b. $HI = 3(HI')$: Lengths in the image are $3$ times the preimage, so $HI' = 3 \times HI$, not $HI = 3(HI')$. False. - c. $3(JK) = J'K'$: Since $J'K' = 3 \times JK$, this is true. - d. $3(m\angle H') = m\angle H$: Angles are equal, so $m\angle H = m\angle H'$. Multiplying by 3 is false. - e. $m\angle J = m\angle J'$: True, angles remain the same. **True statements:** c and e. --- 2. **Problem:** Find image points for $(x,y) \to (6x,6y)$ for points E(-6,3), F(-3,-9), G(0,0). 3. **Apply dilation:** - $E' = (6 \times -6, 6 \times 3) = (-36, 18)$ - $F' = (6 \times -3, 6 \times -9) = (-18, -54)$ - $G' = (6 \times 0, 6 \times 0) = (0,0)$ --- 4. **Problem:** Given $A'(10,4), B'(-2,-6), C'(4,8)$ after a $D_4$ (dilation with scale factor 4) centered at $(0,0)$, find preimage coordinates. 5. **Use inverse dilation:** - $A = \left(\frac{10}{4}, \frac{4}{4}\right) = (2.5, 1)$ - $B = \left(\frac{-2}{4}, \frac{-6}{4}\right) = (-0.5, -1.5)$ - $C = \left(\frac{4}{4}, \frac{8}{4}\right) = (1, 2)$ --- 6. **Problem:** Find scale factor mapping $\triangle EFG \to \triangle E'F'G'$ with $E(-12,6), F(6,18), G(3,-9)$ and $E'(-4,2), F'(2,6), G'(1,-3)$. 7. **Calculate scale factor:** - For $E$: $k = \frac{-4}{-12} = \frac{1}{3}$ - For $F$: $k = \frac{2}{6} = \frac{1}{3}$ - For $G$: $k = \frac{1}{3} = \frac{1}{3}$ All consistent, so scale factor $k = \frac{1}{3}$. --- 8. **Problem:** Identify scale factor and write dilation rule for triangles $PQR$ and $P'Q'R'$ with $P(2,1), Q(4,1), R(3,3)$ and $P'(6,3), Q'(12,3), R'(9,9)$. 9. **Calculate scale factor:** - $k = \frac{6}{2} = 3$ (from $P$ to $P'$) 10. **Dilation rule:** $(x,y) \to (3x, 3y)$ --- 11. **Problem:** Identify scale factor and write dilation rule for quadrilaterals $JKLM$ and $J'K'L'M'$ with $J(-4,-11), K(-7,-13), L(-7,-8), M(-3,-5)$ and $J'(-2,-3), K'(-3,-4), L'(-3,-2), M'(-1,-1)$. 12. **Calculate scale factor:** - For $J$: $k = \frac{-2}{-4} = \frac{1}{2}$ - For $K$: $k = \frac{-3}{-7} \approx 0.4286$ (not consistent) Check $y$ coordinates: - For $J$: $k = \frac{-3}{-11} \approx 0.2727$ (not consistent) Try $x$ and $y$ separately: - $J_x$ scale: $\frac{-2}{-4} = 0.5$ - $J_y$ scale: $\frac{-3}{-11} \approx 0.2727$ Since scale factors differ, dilation is not uniform or center is not origin. 13. **Conclusion:** No single scale factor dilation centered at origin maps $JKLM$ to $J'K'L'M'$. --- **Final answers:** - True statements: c, e - Image points: $E'(-36,18), F'(-18,-54), G'(0,0)$ - Preimage coordinates: $A(2.5,1), B(-0.5,-1.5), C(1,2)$ - Scale factor for $\triangle EFG \to \triangle E'F'G'$: $\frac{1}{3}$ - Scale factor and rule for $\triangle PQR \to \triangle P'Q'R'$: $k=3$, rule $(x,y) \to (3x,3y)$ - No single scale factor dilation for $JKLM \to J'K'L'M'$ centered at origin.