Coordinate Geometry Lecture Notes

Topics Covered:

• Plotting Points
• Gradient (Slope)
• Parallel and Perpendicular Lines
• Distance and Midpoint
• Equation of a Line

Plotting Points

• A point is defined as an ordered pair (x, y).
• x-coordinate: Horizontal distance from the origin.
• y-coordinate: Vertical distance from the origin.
• Example: Points (1,5), (2,10), (3,15) plotted for equation y = 5x.

Key Points:

• Use a graph to extend approximately 5 units on each side of the origin.
• Always use a small "×" to plot points.
• The origin is denoted as (0,0).

Gradient

• Also known as "slope" or "steepness".
• Indicates how steep a line is; higher gradient = steeper line.

Gradient Types:

• Positive: Line rises from left to right.
• Negative: Line falls from left to right.
• Zero: Horizontal line.
• Infinite (undefined): Vertical line.

Parallel Lines

• Never intersect; maintain a constant distance apart.
• Have the same gradient: m1 = m2.

Perpendicular Lines

• Intersect at right angles (90°).
• Product of their gradients is -1: m1 * m2 = -1.*

Gradient Formula

• Calculated as change in y over change in x: [ m = \frac{y_2 - y_1}{x_2 - x_1} ]

Distance/Length of a Line

• Calculate using the distance formula derived from Pythagoras’ Theorem: [ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Worked Example:

• Find distance between points A(3,8) and B(12,3).
• Using formula, distance = ( \sqrt{106} ) = 10.3 units.

Midpoint of a Line

• Midpoint formula: [ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) ]

Equation of a Line

• General form: y = mx + c
  • m = gradient
  • c = y-intercept

Example Problems:

• Find equation given a point and gradient.
• Use substitution into y = mx + c or y - y1 = m(x - x1).

Finding Equation of Perpendicular Line

• Given a line with gradient m, the perpendicular line has a gradient of -1/m.
• Use point-slope form for calculation.

Exercises

• Identify line equations from given options.
• Practice determining line equations and gradients from points and context.

This summary includes key concepts and formulas required for understanding coordinate geometry in the context of plotting points, calculating gradients, and determining line equations.