Section Formula and Mid-Point Formula
302. The Trisection Trick · splitting segments into three equal parts
AP, PQ, and QB are always equal in length.
Trisection divides a segment into three equal parts. The two trisection points of AB divide it in the ratios 1 : 2 and 2 : 1 (from the section formula), so AP = PQ = QB.
What this lesson covers
Try to break it
Drag A and B anywhere. The trisection points P and Q always sit at the 1/3 and 2/3 marks along AB, so AP = PQ = QB exactly. P uses ratio 1:2 → P = (2A + B)/3; Q uses ratio 2:1 → Q = (A + 2B)/3. Swap A and B and the two formulas swap too.
How you build it
Trisect segment AB.
- Mark point A — one end of the segment.
- Mark point B — the other end.
- Draw segment AB.
- Mark point P such that AP : PB = 1 : 2 — that is, P sits one-third of the way from A toward B.
- Mark point Q such that AQ : QB = 2 : 1 — two-thirds of the way from A to B. P and Q trisect segment AB.
The proof, step by step
Prove that the trisection points divide AB into three equal parts.
- Let A = (x₁, y₁) and B = (x₂, y₂).
- P divides AB in ratio 1:2, so P = ((x₁ + 2x₂)/3, (y₁ + 2y₂)/3).
- Q divides AB in ratio 2:1, so Q = ((2x₁ + x₂)/3, (2y₁ + y₂)/3).
- Distance AP = Distance PQ = Distance QB = ⅓ × Distance AB.
Worked example
Find the coordinates of the points of trisection of the line segment joining the points A(6, –2) and B(–8, 10).
Using section formula, P divides AB in 1:2 → P((1(-8)+2(6))/3, (1(10)+2(-2))/3) = (4/3, 2). Q divides AB in 2:1 → Q((2(-8)+1(6))/3, (2(10)+1(-2))/3) = (-10/3, 6).
- (4/3, 2) and (-10/3, 6) — correct
- (4/3, 2) and (-14/3, 6)
- (-2/3, 2) and (-10/3, 6)
- (4/3, 2) and (-10/3, 8)