1. **State the problem:** Calculate the distance $d$ between two points using the distance formula. 2. **Distance formula:** The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate each distance:** - For $d = \sqrt{(1 + 3)^2 + (1 + 2)^2}$: $$d = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$ - For $d = \sqrt{(-2 + 5)^2 + (5 - 2)^2}$: $$d = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$$ - For $d = \sqrt{(4 + 2)^2 + (5 - 3)^2}$: $$d = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}$$ - For $d = \sqrt{(-6 + 1)^2 + (-2 + 5)^2}$: $$d = \sqrt{(-5)^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}$$ - For $d = \sqrt{(6 - 1)^2 + (-4 + 2)^2}$: $$d = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}$$ 4. **Summary of answers:** - $d_1 = 5$ - $d_2 = 3\sqrt{2}$ - $d_3 = 2\sqrt{10}$ - $d_4 = \sqrt{34}$ - $d_5 = \sqrt{29}$ Each distance is calculated by substituting the coordinates into the distance formula and simplifying step-by-step.