1
P\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)
2
M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
3
P\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\right)
FORMULA CHAPTER
Chapter 19 – Internal and External Division of a Straight Line Segment
WBBSEClass 09Mathematics3 formulas
Complete WBBSE Madhyamik Class 9 section formula list covering internal division, midpoint and external division with coordinate examples.
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Internal Section Formula
P\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)
SymbolMeaningUnit
A(x₁, y₁)First endpointcoordinate units
B(x₂, y₂)Second endpointcoordinate units
m : nInternal ratio AP : PBNot applicable
PPoint dividing AB internallyNot applicable
Worked example
Question: Find the point P dividing A(2, 3) and B(8, 9) internally in the ratio 1:2.
Calculation:
Step 1: m = 1, n = 2, A(2, 3), B(8, 9).
Step 2: x = (1 × 8 + 2 × 2) ÷ (1 + 2) = 12 ÷ 3 = 4.
Step 3: y = (1 × 9 + 2 × 3) ÷ (1 + 2) = 15 ÷ 3 = 5.
Answer: P = (4, 5).
Midpoint Formula
M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
SymbolMeaningUnit
A(x₁, y₁)First endpointcoordinate units
B(x₂, y₂)Second endpointcoordinate units
MMidpoint of ABNot applicable
Worked example
Question: Find the midpoint of A(−2, 4) and B(6, 10).
Calculation:
Step 1: x-coordinate = (−2 + 6) ÷ 2 = 2.
Step 2: y-coordinate = (4 + 10) ÷ 2 = 7.
Answer: M = (2, 7).
External Section Formula
P\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\right)
SymbolMeaningUnit
A(x₁, y₁)First endpointcoordinate units
B(x₂, y₂)Second endpointcoordinate units
m : nExternal ratio AP : PBNot applicable
PPoint dividing AB externallyNot applicable
Worked example
Question: Find the point P dividing A(1, 2) and B(5, 6) externally in the ratio 3:1.
Calculation:
Step 1: m = 3, n = 1, A(1, 2), B(5, 6).
Step 2: x = (3 × 5 − 1 × 1) ÷ (3 − 1) = 14 ÷ 2 = 7.
Step 3: y = (3 × 6 − 1 × 2) ÷ (3 − 1) = 16 ÷ 2 = 8.
Answer: P = (7, 8).