1. **State the problem:** We have two similar quadrilaterals ABCD and FGHJ with a ratio of similitude (scale factor) of $7:11$. Given: - $FJ = 11$ - $AB = 2x$, $BC = 3x$, $CD = 4x$, $AD = x$ We need to find the lengths $FG$, $GH$, and $HJ$. 2. **Recall the properties of similar figures:** Corresponding sides of similar figures are proportional to the ratio of similitude. If the ratio of similitude from ABCD to FGHJ is $\frac{7}{11}$, then for any corresponding sides $s_{ABCD}$ and $s_{FGHJ}$: $$\frac{s_{ABCD}}{s_{FGHJ}} = \frac{7}{11}$$ 3. **Identify corresponding sides:** Since $FJ$ corresponds to $AD$, and $FJ = 11$, $AD = x$: $$\frac{x}{11} = \frac{7}{11} \implies x = 7$$ 4. **Calculate the sides of ABCD:** - $AB = 2x = 2 \times 7 = 14$ - $BC = 3x = 3 \times 7 = 21$ - $CD = 4x = 4 \times 7 = 28$ - $AD = x = 7$ 5. **Find corresponding sides in FGHJ:** Using the ratio $\frac{7}{11}$: $$\frac{AB}{FG} = \frac{7}{11} \implies FG = \frac{11}{7} \times AB = \frac{11}{7} \times 14 = 22$$ $$\frac{BC}{GH} = \frac{7}{11} \implies GH = \frac{11}{7} \times BC = \frac{11}{7} \times 21 = 33$$ $$\frac{CD}{HJ} = \frac{7}{11} \implies HJ = \frac{11}{7} \times CD = \frac{11}{7} \times 28 = 44$$ 6. **Final answers:** - $FG = 22$ - $GH = 33$ - $HJ = 44$