1. The problem is to find the distance between two points $A(3,2)$ and $B(4,5)$ in the coordinate plane. 2. The formula to find the distance $d$ between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This formula comes from the Pythagorean theorem, where the distance is the hypotenuse of a right triangle formed by the horizontal and vertical differences between the points. 3. Substitute the coordinates of points $A$ and $B$: $$d = \sqrt{(4 - 3)^2 + (5 - 2)^2}$$ 4. Calculate the differences: $$d = \sqrt{1^2 + 3^2}$$ 5. Square the differences: $$d = \sqrt{1 + 9}$$ 6. Add the squares: $$d = \sqrt{10}$$ 7. The distance between points $A$ and $B$ is: $$\boxed{\sqrt{10}}$$