Coordinate Geometry • Topic 2 of 4

Section Formula

The section formula finds the point that divides a segment joining A(x₁, y₁) and B(x₂, y₂) in a given ratio m:n. For internal division (the point lies between A and B) the coordinates are ((m·x₂+n·x₁)/(m+n), (m·y₂+n·y₁)/(m+n)). The order is the trap: the m goes with the FAR point B and the n with the NEAR point A — get this backwards and you split in ratio n:m instead. Setting m = n = 1 collapses this to the midpoint, so the midpoint is just the 1:1 case. For external division (the point lies on the line but outside the segment) replace the plus signs with minus: ((m·x₂−n·x₁)/(m−n), (m·y₂−n·y₁)/(m−n)). A favourite CAT use is the reverse question — "in what ratio does the x-axis divide the segment?" Put the dividing point’s relevant coordinate to zero and solve for the ratio λ:1; a negative λ means the division is external. The centroid, ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3), is the section point that cuts each median in 2:1 from the vertex.

✅ Solved examples

1. Find the point dividing the segment from A(1, 2) to B(7, 5) internally in ratio 2:1.
x = (2·7 + 1·1)/3 = 15/3 = 5; y = (2·5 + 1·2)/3 = 12/3 = 4. Point is (5, 4).
2. Find the point dividing A(−1, 3) to B(4, −2) externally in ratio 3:2.
x = (3·4 − 2·(−1))/(3−2) = (12+2)/1 = 14; y = (3·(−2) − 2·3)/1 = (−6−6) = −12. Point is (14, −12).
3. In what ratio does the x-axis divide the segment joining (2, −3) and (5, 6)?
Let ratio be λ:1. The y-coordinate of the dividing point is (6λ − 3)/(λ+1) = 0 ⇒ 6λ = 3 ⇒ λ = 1/2. So the ratio is 1:2 (internally, since positive).
4. The centroid of a triangle is (2, 1). Two vertices are (1, 4) and (4, −2). Find the third.
(1+4+x)/3 = 2 ⇒ x = 1; (4−2+y)/3 = 1 ⇒ y = 1. Third vertex is (1, 1).

✏️ Practice — try these, take hints as needed

1. Find the point dividing (0, 0) to (10, 5) internally in ratio 3:2.
x = (3·10 + 2·0)/5.
y = (3·5 + 2·0)/5.
Denominator m+n = 5.
(6, 3)
2. Find the point dividing (2, 1) to (8, 7) internally in ratio 1:2.
m=1 goes with the far point (8,7).
x = (1·8 + 2·2)/3.
y = (1·7 + 2·1)/3.
(4, 3)
3. In what ratio does the y-axis divide the segment from (−3, 4) to (6, −2)?
Set the x-coordinate of the dividing point to 0.
(6λ − 3)/(λ+1) = 0.
Solve for λ:1.
1:2
4. Find the point dividing (−2, −2) to (4, 4) externally in ratio 3:1.
Use the minus-sign (external) formula.
x = (3·4 − 1·(−2))/(3−1).
Denominator m−n = 2.
(7, 7)
5. Two vertices of a triangle are (3, −5) and (−7, 4); its centroid is (2, −1). Find the third vertex.
Sum of x-coords = 3 × centroid x.
3 − 7 + x = 6.
Repeat for y.
(10, −2)

📝 Topic test — 8 questions

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