The Straight Line: Gradient, Equation, Midpoint and Length
Gradient from two points, the equation of a line in several ways, parallel and perpendicular lines, length and midpoint of a segment, and solving simultaneous equations graphically.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
LESSON B9 · EXPLAINER
The Straight Line: Gradient, Equation, Midpoint and Length
Two points are all you need: with them you can find the gradient, the equation, the midpoint, the length — and even the perpendicular bisector.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Explore in 3D
The straight line
Move two points; read the gradient, midpoint and length.
Step 2 · Read
Key notes
Copy these into your notebook in your own words — that is how they stick.
Gradient
m = (y₂ − y₁)/(x₂ − x₁) = rise ÷ run.
Midpoint
((x₁ + x₂)/2, (y₁ + y₂)/2).
Length
√((x₂ − x₁)² + (y₂ − y₁)²).
Equation
y = mx + c; or y − y₁ = m(x − x₁); from two points find m first.
Parallel
Same gradient: m₁ = m₂.
Perpendicular
m₁ × m₂ = −1: negative reciprocal.
Perpendicular bisector
Passes through the midpoint with the perpendicular gradient.
Graphical solution
Intersection of two lines gives the simultaneous solution.
EXAM TIPS & COMMON MISTAKES
Subtract coordinates in the SAME order for numerator and denominator.
Perpendicular gradient: flip AND change the sign — 3/4 becomes −4/3, not 4/3.
Write the final equation in a neat form such as y = mx + c or ax + by = c, as the question asks.
For graph solutions give coordinates and label which line is which.
Play & practise
Games
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Mathematics Arcade.
Step 3 · Practise
Paper 1 practice — multiple choice
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Step 4 · Apply
Paper 2 practice — structured questions
Write full answers, then reveal the mark scheme and tick the points you earned (20 marks in total). Show all your working — method marks count. Your answers are saved on this device.
CSEC Paper 02 style · Module 2 (Coordinate geometry)
Question 1 [12 marks]
A line segment joins the points A(−2, −1) and B(4, 7).
(a) Calculate the midpoint of AB. [2]
MARK SCHEME — (a) (max 2)
(b) Calculate the gradient of AB and the length of AB. [4]
MARK SCHEME — (b) (max 4)
(c) Determine the equation of AB, giving your answer in the form ax + by = c. [3]
MARK SCHEME — (c) (max 3)
(d) Find the equation of the perpendicular bisector of AB. [3]
MARK SCHEME — (d) (max 3)
CSEC Paper 02 style · Module 2 (Lines and graphs)
Question 2 [8 marks]
Two lines are given by y = 2x − 1 and 2y + x = 8.
(a) Find, algebraically, the coordinates of the point where the lines intersect. [3]
MARK SCHEME — (a) (max 3)
(b) Show that the lines are perpendicular. [3]
MARK SCHEME — (b) (max 3)
(c) Write down the equation of the line parallel to y = 2x − 1 passing through the point (0, 4). [2]
MARK SCHEME — (c) (max 2)
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Sources
Where this topic appears in CSEC papers
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.
CSEC syllabus Module 2, SO 3.6 (gradient), 3.7 (equation of a line), 3.8 (parallel and perpendicular gradients), 3.9 (length and midpoint), 3.10 (graphical solution of simultaneous equations)
CSEC P2 January 2025 — Question 4 (midpoint and gradient of CD, equation of the perpendicular bisector, parallel line AB); P2 May/June 2026 — Question 4 (midpoint and gradient of AB)
Straight-line items are a regular part of Paper 01 Relations, Functions and Graphs 1 (four items per paper)