Geometry and Trigonometry - Equation of a straight line

The Gradient ($m$) measures the steepness and direction of a line. Visually, a positive gradient slopes upwards from left to right, while a negative gradient slopes downwards. A horizontal line has a gradient of $0$, and a vertical line has an undefined gradient.

The $y$-intercept ($c$) is the point where the line crosses the vertical $y$-axis. On a graph, this is located at the coordinates $(0, c)$. Conversely, the $x$-intercept is where the line crosses the horizontal $x$-axis, found by setting $y = 0$.

The Gradient-Intercept form, $y = mx + c$, is the most common way to express a straight line. Here, $m$ represents the 'rise over run' (vertical change divided by horizontal change) and $c$ represents the height at which the line hits the $y$-axis.

The Point-Gradient form, $y - y_1 = m(x - x_1)$, is used to find the equation of a line when you know its slope and one specific point $(x_1, y_1)$ that it passes through. This is often rearranged into other forms for the final answer.

Parallel lines are lines that maintain a constant distance from each other and never meet. Visually, they have the same slant, which means their gradients are equal ($m_1 = m_2$).

Perpendicular lines meet at a right angle ($90^{\circ}$). Their gradients are negative reciprocals of each other ($m_1 \times m_2 = -1$). For example, if one line has a gradient of $\frac{2}{3}$, the perpendicular line has a gradient of $-\frac{3}{2}$.

The Midpoint is the exact center point between two coordinates $(x_1, y_1)$ and $(x_2, y_2)$. Visually, it is the 'average' position of the two points on the Cartesian plane.

Distance between two points represents the length of the straight-line segment connecting them. It is calculated by visualizing a right-angled triangle between the points and applying the Pythagorean theorem to the horizontal and vertical distances.