The Core Formulas & Applying Them
Master distance, midpoint, and gradient — plus parallel/perpendicular lines, collinearity, and proving shape properties — with full worked examples.
Analytical Geometry
Table of Contents
Analytical Geometry: Distance, Midpoint & Gradient#
Analytical geometry puts shapes onto the Cartesian ($x$-$y$) plane so you can use algebra to solve geometry problems. In Grade 10, you need three formulas — and you must know when to use each one.
The Three Core Formulas#
Given two points $A(x_1;\, y_1)$ and $B(x_2;\, y_2)$:
| Formula | What it finds | Equation |
|---|---|---|
| Distance | Length between two points | $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ |
| Midpoint | Centre point between two points | $M = \left(\frac{x_1 + x_2}{2};\, \frac{y_1 + y_2}{2}\right)$ |
| Gradient | Steepness of the line | $m = \frac{y_2 - y_1}{x_2 - x_1}$ |
💡 The distance formula is just Pythagoras’ theorem ($a^2 + b^2 = c^2$) on the coordinate plane. The horizontal distance is $a$, the vertical distance is $b$, and the straight-line distance is $c$.
Understanding Gradient#
The gradient ($m$) tells you the direction and steepness of a line:
| Gradient | Line behaviour |
|---|---|
| $m > 0$ | Line slopes upward (from left to right) |
| $m < 0$ | Line slopes downward |
| $m = 0$ | Horizontal line |
| $m$ undefined | Vertical line ($x_2 = x_1$, division by zero) |
Parallel and Perpendicular Lines#
| Relationship | Condition |
|---|---|
| Parallel ($\parallel$) | $m_1 = m_2$ (same steepness) |
| Perpendicular ($\perp$) | $m_1 \times m_2 = -1$ (negative reciprocals) |
Example: If $m_1 = \frac{2}{3}$, the perpendicular gradient is $m_2 = -\frac{3}{2}$.
The Equation of a Straight Line#
Once you know the gradient and a point, you can write the equation:
$$y - y_1 = m(x - x_1) \quad \text{or} \quad y = mx + c$$Deep Dive#
- Core Formulas & Applications — full worked examples for distance, midpoint, gradient, collinear points, and finding equations of lines
🚨 Common Mistakes#
- Sign errors in the distance formula: $(x_2 - x_1)^2$ is always positive (you’re squaring), so the order doesn’t matter. But be careful with negative coordinates.
- Gradient division by zero: If $x_1 = x_2$, the gradient is undefined (vertical line). Don’t write $m = 0$.
- Midpoint is NOT the distance: Midpoint gives you a POINT (coordinates), not a number.
- Perpendicular gradients: The product must be $-1$, not just “the reciprocal”. $m_1 \times m_2 = -1$.
🔗 Related Grade 10 topics:
- Trigonometry — gradient connects to $\tan\theta$ (expanded in Grade 11)
- Functions — the gradient of a straight line is the $a$ in $y = ax + q$
📌 Where this leads in Grade 11: Analytical Geometry: Inclination & Circles — angle of inclination, equation of a circle, tangent lines
⏮️ Euclidean Geometry | 🏠 Back to Grade 10 | ⏭️ Statistics