Summary (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

Analytical geometry combines algebra and geometry to solve problems using coordinates and equations. This summary covers all the essential formulas and concepts you need for your NSC Mathematics exam.

Basic coordinate geometry formulas

These fundamental formulas form the building blocks of analytical geometry. You'll use them constantly throughout your exam.

Theorem of Pythagoras

For any right-angled triangle with sides AC and BC, and hypotenuse AB:

$AB^2 = AC^2 + BC^2$

This theorem helps you find distances between points and is the foundation for the distance formula.

Distance formula

To find the distance between two points A($x_1; y_1$) and B($x_2; y_2$):

$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Gradient (slope)

The gradient measures how steep a line is. For two points A($x_1; y_1$) and B($x_2; y_2$):

$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \text{ or } m_{AB} = \frac{y_1 - y_2}{x_1 - x_2}$

Think "rise over run" - the change in y-values divided by the change in x-values.

Midpoint formula

To find the midpoint M between two points A($x_1; y_1$) and B($x_2; y_2$):

$M(x; y) = \left(\frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2}\right)$

Simply add the coordinates and divide by 2 for each axis.

Collinear points

Three or more points lie on the same straight line when:

$m_{AB} = m_{AM} = m_{MB}$

All gradients between any two points must be equal.

Straight line equations

Different forms of line equations are useful for different situations. Choose the most appropriate form based on the information given.

Two-point form

When you know two points ($x_1; y_1$) and ($x_2; y_2$) on the line:

$\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$

Gradient-point form

When you know the gradient m and one point ($x_1; y_1$):

$y - y_1 = m(x - x_1)$

Gradient-intercept form

When you know the gradient m and y-intercept c:

$y = mx + c$

This is often the most useful form for graphing and analysis.

Special cases

• Horizontal lines: $y = k$ (gradient = 0)
• Vertical lines: $x = k$ (gradient is undefined)

Parallel and perpendicular lines

Understanding the relationship between lines is crucial for many analytical geometry problems.

Parallel lines

Lines are parallel when they have the same gradient and never intersect:

• Same gradients: $m_1 = m_2$
• Same angles with x-axis: $\theta_1 = \theta_2$

Perpendicular lines

Lines are perpendicular when they meet at right angles:

• Product of gradients: $m_1 \times m_2 = -1$
• Angle relationship: $\theta_1 = 90° + \theta_2$

Circle geometry essentials

Inclination of a straight line

The gradient of a line equals the tangent of its angle with the positive x-axis:

$m = \tan \theta \text{ for } 0° \leq \theta < 180°$

Circle equations

Circle with centre at origin (0; 0): $x^2 + y^2 = r^2$

General circle with centre (a; b): $(x - a)^2 + (y - b)^2 = r^2$

Where r is the radius of the circle.

Tangent properties

Worked example: Finding intersection points