1. The problem asks us to find the distance between the points $(1,2)$ and $(4,6)$ using the Pythagorean theorem. 2. The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is derived from the Pythagorean theorem and is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. Substitute the given points into the formula: $$d = \sqrt{(4 - 1)^2 + (6 - 2)^2}$$ 4. Calculate the differences: $$d = \sqrt{3^2 + 4^2}$$ 5. Square the differences: $$d = \sqrt{9 + 16}$$ 6. Add the squares: $$d = \sqrt{25}$$ 7. Take the square root: $$d = 5$$ Therefore, the distance between the points $(1,2)$ and $(4,6)$ is 5 units.