1. **State the problem:** Find the distance between the points $A(5,5)$ and $B(8,1)$ on the Cartesian plane. 2. **Formula used:** The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Apply the formula:** Substitute $x_1=5$, $y_1=5$, $x_2=8$, and $y_2=1$: $$d = \sqrt{(8 - 5)^2 + (1 - 5)^2}$$ 4. **Simplify inside the square root:** $$(8 - 5)^2 = 3^2 = 9$$ $$(1 - 5)^2 = (-4)^2 = 16$$ 5. **Sum the squares:** $$9 + 16 = 25$$ 6. **Take the square root:** $$d = \sqrt{25} = 5$$ **Final answer:** The distance between the points is $5$ units.