Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find the coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, −2).
Step-by-step solution
Idea: Trisection points cut AB into three equal pieces AP = PQ = QB. So P is halfway between A and Q, and Q is halfway between P and B: two midpoint conditions that fix P and Q. Equivalently, each piece is one third of the total shift from A to B.
- Method. AP = PQ = QB, so P is the midpoint of AQ and Q is the midpoint of PB. Let P = (p, r) and Q = (q, s). The midpoint rule gives, for the x-coordinates, 2p = 4 + q and 2q = p + 16.1 mark
- From the first, q = 2p − 4. Substitute: 2(2p − 4) = p + 16, so 4p − 8 = p + 16, 3p = 24, p = 8 and q = 12.1 mark
- y-coordinates: 2r = 7 + s and 2s = r + (−2). So s = 2r − 7, 2(2r − 7) = r − 2, 3r = 12, r = 4 and s = 1.1 mark
- So P = (8, 4) and Q = (12, 1). (Shortcut: the shift from A to B is (12, −9); one third of it is (4, −3). Adding it once to A gives P (8, 4), twice gives Q (12, 1). In general P = ((2x1 + x2) ÷ 3, (2y1 + y2) ÷ 3) and Q = ((x1 + 2x2) ÷ 3, (y1 + 2y2) ÷ 3).)½ mark
- Check: midpoint of A and Q = ((4 + 12) ÷ 2, (7 + 1) ÷ 2) = (8, 4) = P ✓; midpoint of P and B = ((8 + 16) ÷ 2, (4 − 2) ÷ 2) = (12, 1) = Q ✓.½ mark
Check: AP = √(42 + 32) = 5, PQ = 5, QB = 5 and AB = √(122 + 92) = 15 = 3 × 5 ✓.
Answer to write in the exam
AP = PQ = QB ⇒ P is the midpoint of AQ, Q is the midpoint of PB
Let P (p, r), Q (q, s)
2p = 4 + q, 2q = p + 16 ⇒ 3p = 24 ⇒ p = 8, q = 12
2r = 7 + s, 2s = r − 2 ⇒ 3r = 12 ⇒ r = 4, s = 1
∴ P = (8, 4), Q = (12, 1)
Check: midpoint of AQ = (8, 4) = P; midpoint of PB = (12, 1) = Q
Common mistakes that cost marks
- Taking P as the midpoint of AB, (10, 2.5). The midpoint splits AB into two parts, not three.
- Swapping P and Q. P must be the one nearer to A: P (8, 4), not (12, 1).
- Dividing the coordinates of B by 3, giving (16 ÷ 3, −2 ÷ 3). The thirds are of the shift from A to B, which is then added to A.
How this can come in the exam
The points of trisection of the segment joining (0, 0) and (9, 6) are
- (3, 2) and (6, 4)
- (3, 3) and (6, 6)
- (4.5, 3) and (9, 6)
- (2, 3) and (4, 6)
Show answer
(A) (3, 2) and (6, 4)
One third of the shift (9, 6) is (3, 2); add it once and twice to (0, 0).
Find the points of trisection of the segment joining (−2, 5) and (10, −1).
Show answer
Shift = (12, −6) (1 mark); one third = (4, −2) (½ mark). Points (2, 3) and (6, 1) (1 mark). Check: (2, 3) is the midpoint of (−2, 5) and (6, 1) ✓ (½ mark).Try one yourself
Find the points of trisection of the segment joining A (−3, 1) and B (3, 13).
Show answer
Shift (6, 12), one third (2, 4): P (−1, 5) and Q (1, 9).
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