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Collinear

Collinear

Lying on the same line.

 

A straight line with points A, B, and C marked in order, showing three collinear points on the same line.

Worked Example

Problem: Determine whether the points A(1, 2), B(3, 6), and C(5, 10) are collinear.
Step 1: Find the slope from A to B.
mAB=6−23−1=42=2m_{AB} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
Step 2: Find the slope from B to C.
mBC=10−65−3=42=2m_{BC} = \frac{10 - 6}{5 - 3} = \frac{4}{2} = 2
Step 3: Compare the two slopes. Since both slopes equal 2, the three points lie on the same line.
mAB=mBC=2m_{AB} = m_{BC} = 2
Answer: Yes, A, B, and C are collinear because the slope between each consecutive pair of points is the same.

Why It Matters

Collinearity is a key concept in coordinate geometry and proof-writing. Showing that three points are collinear (or not) helps you determine whether a triangle can be formed, verify geometric constructions, and solve problems involving lines and segments.

Common Mistakes

Mistake: Calling two points collinear as if it is a meaningful property.
Correction: Any two distinct points are automatically on the same line. Collinearity is only a useful test for three or more points.

Related Terms

  • Line — The object collinear points share
  • Noncollinear — Points that do not lie on one line
  • Coplanar — Points lying in the same plane
  • Slope — Equal slopes can verify collinearity