308. The Collinearity Check · When three points share a single line

Try to break it

Drag A or B to tilt the line; C slides with it. When A, B, C lie on one line, the slope of AB equals the slope of BC. Pull C off the line and the equality breaks instantly. So the collinearity test boils down to: slope of AB = slope of BC.

How you build it

Place three collinear points.

• Point tool: drop point A anywhere on the canvas.
• Point tool: drop point B away from A — A and B fix the line.
• Line tool: click A, then B — the line AB, extended both ways.
• Point tool: place C anywhere on the line AB. Because C lies on AB, slope BC = slope AB — A, B and C are collinear.

The proof, step by step

Prove that A, B, and C are collinear when AB and BC have equal slopes.

• Given: Points A, B, and C are collinear.
• Since A, B, C lie on the same straight line, they share a common slope, let's call it m.
• The slope of the segment AB is m.
• The slope of the segment BC is also m.
• Therefore, Slope of AB = Slope of BC. Hence proved.

Worked example

Points A(2, 3), B(4, 7), and C(6, k) are collinear. Find the value of k.

Slope of AB = (7-3)/(4-2) = 2. Since A, B, C are collinear, slope of BC must also be 2. So, (k-7)/(6-4) = 2 => k-7 = 4 => k = 11.

• 9
• 10
• 11 — correct
• 12