Ch 1 · 3.1–3.4 Coordinates, Linear Graphs, Gradient, Length and Midpoint · O Level Mathematics 4024

Topic 1 of 2 in Coordinate Geometry — O Level Mathematics 4024 — 48 lessons.

Introduction

Coordinates turn geometry into numbers: a map grid reference, a pixel on a screen, the position of a ship. With them you can draw lines from their equations and measure steepness, distance and the halfway point using simple formulas. (Syllabus sections 3.1–3.4.)

Key ideas — Coordinates

A point (x, y) is x across and y up from the origin; negative values go left or down. The axes divide the plane into four quadrants.

🧑‍🏫 Teacher's explanation

Imagine a city laid out in a perfect grid of streets and avenues. To tell someone exactly where you are, you can't just say "on the corner"; you need to say how many blocks East or West (the x-coordinate) and how many blocks North or South (the y-coordinate) you are from the central town square, which we call the origin (0,0).

📖 Definitions

🔍 Worked example — Easy

Find the midpoint of the line segment joining A(2, 3) and B(6, 7).

🔍 Worked example — Standard

Calculate the length of the line segment connecting P(-1, 2) and Q(3, 5).

🔍 Worked example — Exam-style

Find the gradient of the line passing through (2, -3) and (5, 6).

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Drawing a straight line

Make a table with three x-values (the third is a check), work out y, plot and rule a line across the whole grid. For 3x + 2y = 5, either rearrange to y = 5 - 3x/2 or find the intercepts: x = 0 ⇒ y = 2.5, y = 0 ⇒ x = tfrac53.

🧑‍🏫 Teacher's explanation

Think of a straight line on a graph like a long, perfectly straight mountain road. To draw this road accurately, you don't need to plot every single point along it; you only need a few reliable 'landmarks' to guide your ruler. Once you have two or three points, you can draw a single straight line that passes through all of them.

📖 Definitions

🔍 Worked example — Easy

Draw the graph of y = -2x + 5 by finding three points.

🔍 Worked example — Standard

Find the length and midpoint of the line segment connecting A(1, 2) and B(4, 6).

🔍 Worked example — Exam-style

A line has a gradient of 3 and passes through the point (2, 5). Find the equation of the line in the form y = mx + c.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Gradient

The gradient measures steepness:

🧑‍🏫 Teacher's explanation

Imagine you are walking up a hill. The 'gradient' is simply a number that tells you how steep that hill is. If you walk forward 1 metre and go up 2 metres, the hill is very steep. If you walk forward 10 metres and only go up 1 metre, the hill is quite gentle. In mathematics, we turn this 'steepness' into a number called m.

📖 Definitions

🔍 Worked example — Easy

Find the gradient of the line passing through the points A(2, 3) and B(5, 9).

🔍 Worked example — Standard

Calculate the length of the line segment joining P(-1, 2) and Q(3, 5).

🔍 Worked example — Exam-style

A line segment has endpoints M(4, -2) and N(10, 6). Find the coordinates of the midpoint and the gradient of the line.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Length and midpoint

For A(x_1, y_1) and B(x_2, y_2):

🧑‍🏫 Teacher's explanation

Imagine you are standing on a giant grid-patterned floor. If you want to walk from one tile to another, the 'length' is simply the shortest, straight-line distance between you and your destination. To find this, you can't just count tiles; you have to look at how far you move horizontally (the 'run') and how far you move vertically (the 'rise'). If you draw these movements, they form the two sides of a right-angled triangle, and the straight path is the hypotenuse. This is why we use Pythagoras' theorem to find the length.

Ch 1 · 3.1–3.4 Coordinates, Linear Graphs, Gradient, Length and Midpoint · O Level Mathematics 4024

All chapters in Coordinate Geometry