Summary (Grade 11 NSC Matric Mathematics): Revision Notes

Summary

This revision note covers all the essential formulas and concepts you need to master in analytical geometry. These tools help you work with points, lines, and their relationships on the coordinate plane.

Distance between two points

The distance formula calculates the straight-line distance between any two points on a coordinate plane.

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

Example: Find the distance between points A(2; 3) and B(5; 7).

• $d = \sqrt{(5 - 2)^2 + (7 - 3)^2}$
• $d = \sqrt{3^2 + 4^2}$
• $d = \sqrt{9 + 16} = \sqrt{25} = 5$ units

Gradient of a line

The gradient (or slope) measures how steep a line is and whether it slopes upward or downward.

Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Remember rise over run - the gradient shows how much the line rises (or falls) for every unit it moves horizontally.

Example: Find the gradient of the line through points C(-1; 2) and D(3; 8).

• $m = \frac{8 - 2}{3 - (-1)} = \frac{6}{4} = \frac{3}{2}$

This means the line rises 3 units for every 2 units it moves right.

Mid-point of a line segment

The mid-point is exactly halfway between two points on a line segment.

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

You simply average the x-coordinates and average the y-coordinates separately.

Example: Find the mid-point of the line segment joining E(4; -2) and F(-2; 6).

• $M = \left(\frac{4 + (-2)}{2}, \frac{-2 + 6}{2}\right)$
• $M = \left(\frac{2}{2}, \frac{4}{2}\right) = (1; 2)$

Parallel and perpendicular lines

Understanding the relationships between lines is crucial for solving analytical geometry problems.

Example: If line 1 has gradient $\frac{2}{3}$, then:

• A parallel line also has gradient $\frac{2}{3}$
• A perpendicular line has gradient $-\frac{3}{2}$

Forms of straight line equations

There are several ways to write the equation of a straight line, each useful in different situations.

General form

Formula: $ax + by + c = 0$

This is useful for representing any line, including vertical lines.

Two-point form

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

Use this when you know two points on the line.

Gradient-point form

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

Use this when you know the gradient and one point on the line.

Gradient-intercept form (standard form)

Formula: $y = mx + c$

This is the most common form, where $m$ is the gradient and $c$ is the y-intercept.

Angle of inclination

The angle of inclination ($\theta$) is the angle between a line and the positive x-axis, measured anticlockwise.

Formula: $m = \tan \theta$

This connects the gradient to the angle the line makes with the x-axis.

Example: If a line has an angle of inclination of 45°:

• $m = \tan 45° = 1$