Chapter 13 · ICSE Class 10 Mathematics (Selina Concise)

Section & Mid-Point Formula

Find the coordinates of a point dividing a line segment in a given ratio, and find midpoints of segments.

📍 Chapter 13

The section formula showing a point dividing a line segment in the ratio 1 to 2 A line segment from point A(2,3) to point B(8,9), with point P dividing it internally in the ratio 1:2, located at (4,5) A(2,3) B(8,9) P(4,5) 1 2 P divides AB in the ratio 1 : 2
13.1

The Section Formula

A point P dividing the segment joining (x₁,y₁) and (x₂,y₂) in the ratio m:n has coordinates ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

1

Find the point dividing the segment joining (2,3) and (8,9) in the ratio 1:2.

x=(1×8+2×2)/3=12/3=4; y=(1×9+2×3)/3=15/3=5; point is (4,5)

2

Find the point dividing the segment joining (0,0) and (10,15) in the ratio 2:3.

x=(2×10+3×0)/5=4; y=(2×15+3×0)/5=6; point is (4,6)

13.2

The Mid-point Formula

The midpoint of a segment joining (x₁,y₁) and (x₂,y₂) is ((x₁+x₂)/2, (y₁+y₂)/2) — the special case where m:n=1:1.

1

Find the midpoint of (4,6) and (10,14).

((4+10)/2, (6+14)/2)=(7,10)

2

If (5,7) is the midpoint of A(2,3) and B, find B.

5=(2+x)/2 → x=8; 7=(3+y)/2 → y=11; B=(8,11)

Practice

Exercises

1. Find the point dividing the segment joining (1,2) and (7,8) in the ratio 1:1.

2. Find the point dividing the segment joining (0,0) and (6,9) in the ratio 2:1.

3. Find the midpoint of (−2,5) and (6,−3).

4. If (3,4) is the midpoint of A(1,2) and B, find B.

5. Find the point dividing the segment joining (−4,0) and (0,8) in the ratio 3:1.

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