Three points are collinear if they all lie on one straight line. The test uses gradients: if the gradient from A to B equals the gradient from B to C, the points are collinear. A related question asks for a missing coordinate that makes three points line up.
Why does equal gradient prove it?
The gradient of a straight line is the same everywhere along it. If AB and BC have equal gradients, they point in the same direction. Since they also share the point B, they cannot be two separate lines, so A, B and C sit on one line.
Finding each gradient is covered in calculating a gradient from two points. This lesson uses that skill twice in one question.
How do you work it out, step by step?
- Choose a pair of segments that share a point, such as AB and BC.
- Find each gradient with (y₂ − y₁)/(x₂ − x₁), keeping the same order for top and bottom.
- Simplify both fractions fully.
- Compare. Equal gradients mean collinear; different gradients mean not collinear.
- Write a conclusion that names the common point.
Worked example
Show that A(1, 2), B(3, 6) and C(6, 12) are collinear.
Gradient of AB: (6 − 2)/(3 − 1) = 4/2 = 2.
Gradient of BC: (12 − 6)/(6 − 3) = 6/3 = 2.
Conclusion: both gradients equal 2, and B is common to AB and BC. Therefore A, B and C are collinear.
Independent check: the line through A with gradient 2 is y − 2 = 2(x − 1), so y = 2x. Test the other points: B gives 2 × 3 = 6 ✓ and C gives 2 × 6 = 12 ✓. All three points satisfy y = 2x.
Contrast case: P(0, 1), Q(2, 5) and R(5, 10). The gradient of PQ is 4/2 = 2, but the gradient of QR is 5/3. They differ, so P, Q and R are not collinear.
The mistake to watch for
The common slip is turning one of the gradients upside down.
Mistaken working: gradient of BC = (6 − 3)/(12 − 6) = 3/6 = 1/2, so “the points are not collinear”.
The student put the x-difference on top and the y-difference on the bottom for BC only.
The gradient is always change in y over change in x. Write the word “rise over run” next to the working if needed. The correct BC gradient is 6/3 = 2, which matches AB.
A second slip is comparing AB with AC and forgetting the order of subtraction, which can flip a sign and give a false “not collinear”.
Check yourself
Try these without a calculator, then open each answer.
1. Are (−1, −5), (2, 1) and (5, 7) collinear?
Show answer
Gradient of the first two points: (1 − (−5))/(2 − (−1)) = 6/3 = 2. Gradient of the last two points: (7 − 1)/(5 − 2) = 6/3 = 2. The gradients are equal and (2, 1) is shared, so yes, collinear.
2. Are (0, 3), (4, 5) and (8, 8) collinear?
Show answer
Gradient of the first two points: (5 − 3)/(4 − 0) = 2/4 = 1/2. Gradient of the last two points: (8 − 5)/(8 − 4) = 3/4. The gradients differ, so no, not collinear.
3. Find k so that (1, 3), (3, 7) and (5, k) are collinear.
Show answer
Gradient of the first two points: (7 − 3)/(3 − 1) = 2. For the next segment: (k − 7)/(5 − 3) = 2, so k − 7 = 4 and k = 11. Check: the gradient from (1, 3) to (5, 11) is 8/4 = 2. ✓
Where this leads next
You have now met every skill in this module. Go back through identifying parallel and perpendicular lines if you want to compare how gradients decide each case, then work through the coordinate geometry practice set. The non-calculator working trainer is useful for keeping fractions exact.
Students who lose marks on written conclusions often know the maths already. A teacher in online one-to-one Mathematics tuition can show how to lay out the reasoning so the examiner sees every step.