LEARNING OBJECTIVES
What you will be able to do
- plot and interpret Cartesian coordinates
- draw linear graphs from equations
- calculate gradients, lengths and midpoints
- find equations of straight lines
- use gradient relationships for parallel and perpendicular lines
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Describe straight lines precisely using position, distance, gradient and equations.
Coordinate geometry turns a diagram into algebra. A point records position, while a line equation describes every point on that line.
The gradient is the link between the visual and algebraic views: it measures steepness on the graph and appears as m in y = mx + c.
SECTION 01
Coordinates and drawing straight lines
A point (x, y) is located by moving horizontally to x and vertically to y. An equation generates coordinate pairs that all lie on one graph.
For an equation not already in y = mx + c form, rearrange it or construct a table. Label axes, state the scale and plot at least three points so one error can be detected.
DETAILED EXPLANATION
- y = k is a horizontal line.
- Two accurate points determine a straight line, but a third point helps check errors.
| Form | What it shows |
|---|---|
| y = mx + c | gradient m and y-intercept c |
| ax + by = c | rearrange to identify gradient and intercept |
| x = k | vertical line through x = k; gradient undefined |
| y = k | horizontal line through y = k; gradient 0 |
Drawing 3x + 2y = 12
- Rearrange: 2y = 12 − 3x, so y = 6 − 1.5x.
- When x = 0, y = 6.
- When y = 0, x = 4.
- Plot (0, 6) and (4, 0), then join them.
Answer: The graph has gradient −1.5 and y-intercept 6.
SECTION 02
Gradient, length and midpoint
Gradient compares vertical change with horizontal change. Read both changes in the same direction so their signs are consistent.
The length formula comes from Pythagoras. The midpoint is found by averaging corresponding coordinates.
DETAILED EXPLANATION
- A horizontal line has gradient 0. A vertical line has undefined gradient.
Line segment from A(−2, 5) to B(6, −1)
- Gradient = (−1 − 5)/(6 − (−2)) = −6/8 = −3/4.
- Length = √(8² + (−6)²) = √100 = 10.
- Midpoint = ((−2 + 6)/2, (5 − 1)/2).
Answer: Gradient = −3/4, length = 10 and midpoint = (2, 2).
SECTION 03
Equations of lines
In y = mx + c, m is the gradient and c is the y-coordinate where the line crosses the vertical axis.
Lines may also be written as ax + by = c or x = k. Rearrange to y = mx + c when you need to read the gradient and y-intercept.
If a gradient and one point are known, substitute them into y = mx + c to find c, or use point-gradient form.
Equation through (3, −2) with gradient 4
- Begin with y = 4x + c.
- Substitute x = 3 and y = −2: −2 = 12 + c.
- Therefore c = −14.
Answer: The equation is y = 4x − 14.
SECTION 04
Parallel and perpendicular lines
Parallel lines have equal gradients. Perpendicular non-vertical lines have gradients whose product is −1. Horizontal and vertical lines form the special perpendicular pair where one gradient is 0 and the other is undefined.
A perpendicular bisector passes through a segment's midpoint and has gradient equal to the negative reciprocal of the segment's gradient.
Perpendicular bisector of A(−1, 2) and B(5, 6)
- Midpoint = (2, 4).
- Gradient AB = (6 − 2)/(5 − (−1)) = 2/3.
- Perpendicular gradient = −3/2.
- Use y − 4 = −3/2(x − 2).
Answer: The perpendicular bisector is y = −3x/2 + 7.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Coordinates describe position.
- Gradient is vertical change divided by horizontal change.
- Distance uses Pythagoras and midpoint uses coordinate averages.
- A line equation records gradient and intercept.
- Parallel gradients match; perpendicular gradients multiply to −1.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- plot and interpret Cartesian coordinates
- draw linear graphs from equations
- calculate gradients, lengths and midpoints
- find equations of straight lines
- use gradient relationships for parallel and perpendicular lines