1. Distance Formula
The distance between any two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ in the Cartesian plane:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
- Distance of point $P(x, y)$ from the origin $O(0, 0)$: $OP = \sqrt{x^2 + y^2}$.
2. Section Formula (Internal Division)
The coordinates of the point $P(x, y)$ dividing the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in ratio $m_1 : m_2$:
$$x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \quad y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$$
Mid-Point Formula:
When the point divides the line segment equally ($1:1$ ratio):
$$M(x, y) = \left(\frac{x_1 + x_2}{2}, \, \frac{y_1 + y_2}{2}\right)$$
