Find the coordinates of the points of trisection of the segment joining A(−3, 6) and B(6, −3).

1. 1 · Write what is given and what is to be found. Given: A(−3, 6) and B(6, −3). To find: P and Q on AB with AP = PQ = QB.
2. 2 · Draw the figure. Draw segment AB with P and Q marking three equal parts. Write 1 : 2 for P and 2 : 1 for Q.
3. 3 · Write the formula or theorem. Section formula: ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂))
4. 4 · Find P (ratio 1 : 2).
   • x = (1 × 6 + 2 × (−3))/3 = 0/3 = 0
   • y = (1 × (−3) + 2 × 6)/3 = 9/3 = 3
   • P is (0, 3)
5. 5 · Find Q (ratio 2 : 1).
   • x = (2 × 6 + 1 × (−3))/3 = 9/3 = 3
   • y = (2 × (−3) + 1 × 6)/3 = 0/3 = 0
   • Q is (3, 0)
6. 6 · Check and box the answer. Each step from A to P, P to Q and Q to B adds (3, −3), so the parts are equal. Answer: P(0, 3) and Q(3, 0).

Scoring tip: Write the ratio before each point. Q is also the mid-point of PB, which gives a quick check.