1. The problem is to explain the concept of length between two points $p$ and $q$ in a coordinate system. 2. The length (or distance) between two points $p=(x_1,y_1)$ and $q=(x_2,y_2)$ in a 2D plane is given by the distance formula derived from the Pythagorean theorem: $$\text{Length} = d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. This formula calculates the straight-line distance between the points by finding the difference in their $x$-coordinates and $y$-coordinates, squaring these differences, summing them, and then taking the square root. 4. Important rules: - The length is always non-negative. - If $p$ and $q$ are the same point, the length is zero. 5. Example: If $p=(1,2)$ and $q=(4,6)$, then $$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 6. This means the length between points $p$ and $q$ is 5 units.