Points $A(x_1, y_1)$ and $B(x_2, y_2)$: $$\text{Gradient } m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta$$ $$\text{Distance } AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad \text{Midpoint} = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$$

Equation of a straight line

$$y = mx + c \qquad y - y_1 = m(x - x_1) \qquad \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \qquad \frac{x}{a} + \frac{y}{b} = 1$$

• Horizontal line: $y = k$ (gradient $0$). Vertical line: $x = k$ (gradient undefined).
• For $ax + by + c = 0$: gradient $= -\dfrac{a}{b}$, $x$-intercept $= -\dfrac{c}{a}$, $y$-intercept $= -\dfrac{c}{b}$.

Parallel and perpendicular lines

• Parallel: $m_1 = m_2$.
• Perpendicular: $m_1 m_2 = -1$, i.e. $m_2 = -\dfrac{1}{m_1}$.

Simultaneous equations

Solve by elimination or substitution; graphically the solution is the intersection point of the two lines.

Section formula

Point dividing $AB$ in the ratio $m:n$ (from $A$): $$\left(\frac{n x_1 + m x_2}{m + n},\ \frac{n y_1 + m y_2}{m + n}\right)$$