Revision (Grade 12 NSC Matric Mathematics): Revision Notes

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Essential formulas for straight lines

Understanding the fundamental formulas is crucial for solving analytical geometry problems. These formulas form the foundation for all straight line calculations.

Distance formula

The distance formula calculates the length between any two points on a coordinate plane. For points A(x₁; y₁) and B(x₂; y₂):

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

Gradient formula

The gradient (or slope) measures how steep a line is. It represents the rate of change between two points:

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

The gradient tells us:

• How much y increases for every unit increase in x
• Whether the line slopes upward (positive gradient) or downward (negative gradient)

Midpoint formula

The midpoint is the exact centre point between two given points:

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

This formula simply averages the x-coordinates and y-coordinates separately.

Forms of straight line equations

There are three main forms of straight line equations, each useful in different situations.

Two-point form

When you know two specific points on the line, use:

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

This form directly uses the coordinates of both points to establish the line's equation.

Gradient-point form

When you know the gradient and one point on the line:

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

Gradient-intercept form

When you know the gradient and y-intercept:

$y = mx + c$

where:

• $m$ is the gradient
• $c$ is the y-intercept (where the line crosses the y-axis)

Special cases: horizontal and vertical lines

Some lines have special properties that make their equations simpler.

Horizontal lines

Horizontal lines have zero gradient and are parallel to the x-axis. Their equation is:

$y = k$

where $k$ is a constant representing the y-coordinate of every point on the line.

Vertical lines

Vertical lines have undefined gradient and are parallel to the y-axis. Their equation is:

$x = k$

where $k$ is a constant representing the x-coordinate of every point on the line.

Worked example: quadrilateral properties

Inclination of a line

The angle of inclination is the angle θ that a line makes with the positive x-axis, measured anticlockwise.

Key relationship

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

This connects the gradient of a line directly to its angle of inclination.

Lines with positive gradients

When $m > 0$, the line slopes upward from left to right, and θ is an acute angle (0° < θ < 90°).

Example: For $m = 1.2$

• $\tan \theta = 1.2$
• $\theta = \tan^{-1}(1.2) = 50.2°$

Lines with negative gradients

When $m < 0$, the line slopes downward from left to right, and θ is an obtuse angle (90° < θ < 180°).

Example: For $m = -0.7$

• $\tan \theta = -0.7$
• Reference angle = $\tan^{-1}(0.7) = 35°$
• Obtuse angle = $-35° + 180° = 145°$

Worked example: finding inclination angles

Parallel and perpendicular lines

Understanding the relationship between gradients helps identify parallel and perpendicular lines.

Parallel lines

Parallel lines have identical gradients and will never intersect.

Condition: $m_1 = m_2$

The lines also have equal angles of inclination: $\theta_1 = \theta_2$

Perpendicular lines

Perpendicular lines intersect at right angles (90°).

Condition: $m_1 \times m_2 = -1$

The angles of inclination are related by: $\theta_1 = 90° + \theta_2$

Worked example: parallel and perpendicular lines