7.3 Partitioning a Directed Line Segment

Partitioning a Directed Line Segment

Standard GEO-G.GPE.6 asks you to find the point that divides a directed line segment in a given ratio. "Directed" means the segment has a starting point and an ending point — going from A to B is different from going from B to A, so always read which point comes first.

The Section Formula

To find the point P that partitions the segment from A to B in the ratio m:n (so that AP:PB = m:n), use:

P = A + (m/(m + n))(B − A), applied separately to the x- and the y-coordinate.

The fraction t = m/(m + n) is the fraction of the way from A to B. The single most common Regents error is using m/n instead of m/(m + n): a ratio of 1:2 means P is 1/3 of the way from A to B, not 1/2. Think of it as a weighted average; an equivalent form is:

P = ((n·xₐ + m·x_b)/(m + n), (n·yₐ + m·y_b)/(m + n)).

Fraction-of-the-way reference

Notice the midpoint is just the 1:1 case, where t = 1/2 — the section formula reduces exactly to the midpoint formula from Section 7.1.

Worked Examples

Example 1 (ratio 1:2): Point P divides the segment from A(2, 4) to B(8, 10) so that AP:PB = 1:2. Here t = 1/(1 + 2) = 1/3. The change from A to B is (B − A) = (8 − 2, 10 − 4) = (6, 6). One third of that change is (2, 2). So P = (2 + 2, 4 + 2) = (4, 6).

Example 2 (ratio 2:3): Point P divides the segment from A(−1, 4) to B(9, −6) so that AP:PB = 2:3. Here t = 2/(2 + 3) = 2/5. The vector from A to B is (10, −10). Two fifths of it is (4, −4). So P = (−1 + 4, 4 + (−4)) = (3, 0).

Example 3 (midpoint as 1:1): Partition A(2, 3) to B(8, 11) in ratio 1:1. Then t = 1/2, and P = (2 + ½·6, 3 + ½·8) = (5, 7) — exactly the midpoint. This confirms the section formula generalizes the midpoint.

Example 4 (ratio 3:1): Partition A(−4, 5) to B(4, −7) so that AP:PB = 3:1. The fraction from A is 3/(3 + 1) = 3/4. The change A to B is (8, −12), and 3/4 of it is (6, −9), so P = (−4 + 6, 5 + (−9)) = (2, −4). Because the first ratio number is larger, P sits closer to B, as expected.

The Weighted-Average Form in Action

The equivalent form P = ((n·xₐ + m·x_b)/(m + n), (n·yₐ + m·y_b)/(m + n)) reaches the same answer without a displacement step. Redo Example 1 (A(2, 4), B(8, 10), ratio 1:2, so m = 1 and n = 2): x = (2·2 + 1·8)/3 = 12/3 = 4 and y = (2·4 + 1·10)/3 = 18/3 = 6, giving (4, 6). The larger weight multiplies the closer endpoint, which is why n pairs with A.

Direction Matters

Because the segment is directed, always start from the first-named endpoint. Partitioning A→B in the ratio 1:3 lands 1/4 of the way from A, but partitioning B→A in 1:3 lands 1/4 of the way from B, which is 3/4 of the way from A — a completely different point. If a problem says "the point that divides AB such that AP:PB = 3:1," A is the start and P sits three quarters of the way toward B.

Reading Regents Phrasing and Checking Your Answer

Regents items phrase this task several ways: "the point that divides," "partitions the segment," or "located m/(m+n) of the distance from A to B." All describe the same computation. When the wording gives a fraction directly — for example, "P is 2/5 of the way from A to B" — that fraction is t, so you jump straight to P = A + t(B − A). After finding P, verify it lies on the segment: its coordinates should fall between A and B, and the direction from A to P should match the direction from A to B. If P lands outside the segment or on the wrong side, you probably used m/n or started from the wrong endpoint.

Common traps

• Using m/n instead of m/(m + n) — a 1:3 ratio is 1/4 of the way, not 1/3.
• Starting from the wrong endpoint when the ratio is not symmetric.
• Reversing the ratio, so AP:PB = 2:3 gets computed as 3:2.
• Forgetting to add the fractional change back to point A; the fraction of the vector is only the displacement, not the final coordinates.