8 min readConcept sheet1-mark MCQ, 2-mark and 3-mark questions; often a part of the case-based question

Section formula and mid-point

The section formula finds the point dividing a segment in a ratio m₁ : m₂. The mid-point is the case m₁ = m₂.

Section formula (internal division)P = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂))

Mid-point formula((x₁ + x₂)/2, (y₁ + y₂)/2)

Matching rulem₁ multiplies the far end B(x₂, y₂); m₂ multiplies A(x₁, y₁)

Points of trisectionThe points dividing AB in 1 : 2 and 2 : 1

What the ratio means

Let P lie on the segment AB. If AP : PB = m₁ : m₂, we say P divides AB internally in the ratio m₁ : m₂. The section formula gives the coordinates of P. Notice the cross pattern: m₁ multiplies B's coordinates and m₂ multiplies A's. The formula comes from similar triangles (AA), drawn with horizontal and vertical lines through A, P and B.

Example 1: a point dividing in a given ratio

Problem

Find the point that divides the join of A(−2, 5) and B(8, −5) in the ratio 3 : 2.

Working

x₁ = −2, y₁ = 5, x₂ = 8, y₂ = −5, m₁ = 3, m₂ = 2
Formula: x = (m₁x₂ + m₂x₁)/(m₁ + m₂) and y = (m₁y₂ + m₂y₁)/(m₁ + m₂)
x = (3 × 8 + 2 × (−2))/(3 + 2) = (24 − 4)/5 = 20/5 = 4
y = (3 × (−5) + 2 × 5)/(3 + 2) = (−15 + 10)/5 = −5/5 = −1

Answer

The point is (4, −1).

Example 2: points of trisection

Problem

Find the points of trisection of the segment joining A(1, −2) and B(−5, 7).

Working

The points of trisection P and Q cut AB into three equal parts: AP = PQ = QB.
P divides AB in 1 : 2: x = (1 × (−5) + 2 × 1)/3 = −3/3 = −1
y = (1 × 7 + 2 × (−2))/3 = 3/3 = 1, so P is (−1, 1)
Q divides AB in 2 : 1: x = (2 × (−5) + 1 × 1)/3 = −9/3 = −3
y = (2 × 7 + 1 × (−2))/3 = 12/3 = 4, so Q is (−3, 4)

Answer

P(−1, 1) and Q(−3, 4)

Example 3: the missing end from a mid-point

Problem

M(2, −1) is the mid-point of AB, and A is (−3, 4). Find B.

Working

Mid-point formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Let B be (x, y). Then (−3 + x)/2 = 2 and (4 + y)/2 = −1.
−3 + x = 4, so x = 7
4 + y = −2, so y = −6

Answer

B is (7, −6).

How to draw the figure for section problems

Draw a straight segment and label its ends A(x₁, y₁) and B(x₂, y₂).
Mark P on it, nearer to A if m₁ is smaller than m₂. Write m₁ over AP and m₂ over PB.
For trisection, mark P and Q so that the segment looks cut into three equal parts.
You do not need graph paper. A rough, labelled figure is enough.

In real life

Friends who meet halfway between their homes are using the mid-point formula. A gardener planting a sapling one-third of the way along a path uses the section formula.

Exam angle

1 marks: “The mid-point of the segment joining (−4, 6) and (8, −2) is:” ((−4 + 8)/2, (6 − 2)/2) = (2, 2).

2 marks: “Find the point that divides the join of (4, −3) and (−6, 7) in the ratio 3 : 2.” x = (3 × (−6) + 2 × 4)/5 = −10/5 = −2 and y = (3 × 7 + 2 × (−3))/5 = 15/5 = 3. The point is (−2, 3).

3 marks: “Find the points that divide the segment joining A(−4, 0) and B(4, 8) into four equal parts.” The mid-point of AB is M(0, 4). The mid-point of AM is (−2, 2) and the mid-point of MB is (2, 6). The points are (−2, 2), (0, 4) and (2, 6).

Remember it

Cross and add: m₁ with the far end, m₂ with the near end, then divide by m₁ + m₂.