1. **Stating the problem:** We are given points F(10,12), D(26,10), and E(18,0) as part of a polygonal chain with points A, B, C, D, E, and F connected in a specific sequence forming overlapping rhombus shapes. 2. **Understanding the problem:** To analyze or solve problems related to these points, such as finding distances, slopes, or verifying shapes like rhombuses, we use coordinate geometry formulas. 3. **Distance formula:** The distance between two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 4. **Calculate distances between given points:** - Distance $FD$: $$FD = \sqrt{(26 - 10)^2 + (10 - 12)^2} = \sqrt{16^2 + (-2)^2} = \sqrt{256 + 4} = \sqrt{260} = 2\sqrt{65}$$ - Distance $DE$: $$DE = \sqrt{(18 - 26)^2 + (0 - 10)^2} = \sqrt{(-8)^2 + (-10)^2} = \sqrt{64 + 100} = \sqrt{164} = 2\sqrt{41}$$ 5. **Slope formula:** The slope of the line through points $P_1$ and $P_2$ is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 6. **Calculate slopes to check parallelism or perpendicularity:** - Slope $FD$: $$m_{FD} = \frac{10 - 12}{26 - 10} = \frac{-2}{16} = -\frac{1}{8}$$ - Slope $DE$: $$m_{DE} = \frac{0 - 10}{18 - 26} = \frac{-10}{-8} = \frac{5}{4}$$ 7. **Interpretation:** These calculations help verify the shape properties of the polygonal chain and the rhombus formed by points including F, D, and E. **Final answer:** - Distance $FD = 2\sqrt{65}$ - Distance $DE = 2\sqrt{41}$ - Slope $FD = -\frac{1}{8}$ - Slope $DE = \frac{5}{4}$