1. **Problem statement:** We have two similar rhombuses ABCD and ECFA. The diagonals EF and BD measure 4 m and 16 m respectively. We need to find the perimeter of the larger rhombus ABCD, rounded to 1 decimal place. 2. **Key properties and formula:** - Rhombuses have all sides equal. - The diagonals of a rhombus bisect each other at right angles. - Since ABCD and ECFA are similar, the ratio of their corresponding sides equals the ratio of their corresponding diagonals. 3. **Step 1: Find the scale factor between the rhombuses.** The diagonal BD of ABCD corresponds to diagonal EF of ECFA. Scale factor $k = \frac{BD}{EF} = \frac{16}{4} = 4$ 4. **Step 2: Calculate the side length of the smaller rhombus ECFA.** Since the diagonals of a rhombus are perpendicular and bisect each other, the side length $s$ can be found using half the diagonals: Half of diagonal EF = $\frac{4}{2} = 2$ Half of diagonal EC (unknown, but since ECFA is similar to ABCD, and EF corresponds to BD, the other diagonal of ECFA corresponds to AC of ABCD, but we don't need it explicitly here because we can find side length directly from EF and the scale factor.) However, since we only have EF and BD, and the scale factor is 4, the side length of ABCD is 4 times the side length of ECFA. 5. **Step 3: Calculate side length of ABCD using diagonal BD.** For rhombus ABCD, half diagonals are $\frac{BD}{2} = 8$ and $\frac{AC}{2} = x$ (unknown). Side length $s = \sqrt{8^2 + x^2} = \sqrt{64 + x^2}$ Similarly, for ECFA, half diagonals are $\frac{EF}{2} = 2$ and $\frac{CA}{2} = \frac{x}{4}$ (since scale factor is 4). Side length of ECFA $= \sqrt{2^2 + \left(\frac{x}{4}\right)^2} = \sqrt{4 + \frac{x^2}{16}}$ Since side length of ABCD is 4 times side length of ECFA: $$\sqrt{64 + x^2} = 4 \times \sqrt{4 + \frac{x^2}{16}}$$ 6. **Step 4: Solve for $x$.** Square both sides: $$64 + x^2 = 16 \times \left(4 + \frac{x^2}{16}\right) = 64 + x^2$$ This simplifies to an identity, meaning $x$ can be any value, so the problem is consistent. 7. **Step 5: Calculate side length of ECFA using given diagonal EF = 4 m.** Since ECFA is a rhombus, its side length $s_{ECFA} = \frac{EF}{\sin \theta}$ where $\theta$ is the angle between diagonals. But we don't have $\theta$. Alternatively, use the fact that the diagonals bisect at right angles, so side length $s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}$. We know $d_1 = EF = 4$, but $d_2$ of ECFA is unknown. 8. **Step 6: Use scale factor to find $d_2$ of ECFA.** Since scale factor $k=4$, and $d_2$ of ABCD is $BD=16$, the corresponding diagonal of ECFA is $\frac{16}{4} = 4$ m. So both diagonals of ECFA are 4 m. 9. **Step 7: Calculate side length of ECFA.** $$s_{ECFA} = \sqrt{\left(\frac{4}{2}\right)^2 + \left(\frac{4}{2}\right)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828$$ 10. **Step 8: Calculate side length of ABCD.** $$s_{ABCD} = k \times s_{ECFA} = 4 \times 2.828 = 11.3137$$ 11. **Step 9: Calculate perimeter of ABCD.** Perimeter $P = 4 \times s_{ABCD} = 4 \times 11.3137 = 45.2548$ 12. **Step 10: Round to 1 decimal place.** $$P \approx 45.3$$ **Answer:** The perimeter of rhombus ABCD is approximately 45.3 m. (Note: The given options do not include 45.3 m, so please verify the problem data or options.)