AI generatedGeometry and measuresGCSE Higher

Proving points are collinear

Combine scalar-multiple and common-point reasoning.

  • Find two vectors sharing one point, such as AB and AC.
  • Show one is a single scalar multiple of the other, using the same scalar for every component.
  • State both facts: parallel vectors and a common point. Therefore the three points are collinear.

Example 1

Prove that A, B and C are collinear.

Points on the same straight lineABCSketch only: use the vectors to prove collinearity.
Factor out the common scalar.The vectors are parallel and both start at A.
A, B and C are collinear: parallel vectors with the common point A lie on the same straight line.

Example 2

Prove that A, B and C are collinear.

Points on the same straight lineABCSketch only: use the vectors to prove collinearity.
Factor out the common scalar.The vectors are parallel and both start at A.
A, B and C are collinear: parallel vectors with the common point A lie on the same straight line.

Your turn

Question 1

Prove that A, B and C are collinear.

Points on the same straight lineABCSketch only: use the vectors to prove collinearity.
Check answer
Factor out the common scalar.The vectors are parallel and both start at A.
A, B and C are collinear: parallel vectors with the common point A lie on the same straight line.
Question 2

Prove that A, B and C are collinear.

Points on the same straight lineABCSketch only: use the vectors to prove collinearity.
Check answer
Factor out the common scalar.The vectors are parallel and both start at A.
A, B and C are collinear: parallel vectors with the common point A lie on the same straight line.
Question 3

Prove that A, B and C are collinear.

Points on the same straight lineABCSketch only: use the vectors to prove collinearity.
Check answer
Factor out the common scalar.The vectors are parallel and both start at A.
A, B and C are collinear: parallel vectors with the common point A lie on the same straight line.
Question 4

Prove that A, B and C are collinear.

Points on the same straight lineABCSketch only: use the vectors to prove collinearity.
Check answer
Factor out the common scalar.The vectors are parallel and both start at A.
A, B and C are collinear: parallel vectors with the common point A lie on the same straight line.
Question 5

Prove that A, B and C are collinear.

Points on the same straight lineABCSketch only: use the vectors to prove collinearity.
Check answer
Factor out the common scalar.The vectors are parallel and both start at A.
A, B and C are collinear: parallel vectors with the common point A lie on the same straight line.
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