📖 Chapter 7 — Coordinate Geometry (CBSE Class 10 Maths): Detailed notes — the distance formula and the section formula, with worked steps.
Chapter 7 — Coordinate Geometry
📍 1. Distance Formula
The distance between two points P(x₁, y₁) and Q(x₂, y₂) is:
PQ = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
Distance of a point (x, y) from the origin = √(x² + y²).
✂️ 2. Section Formula (Internal Division)
The point P that divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m : n is:
P = ( (mx₂ + nx₁)/(m+n) , (my₂ + ny₁)/(m+n) )
⚖️ 3. Mid-point Formula
Mid-point of A(x₁,y₁) and B(x₂,y₂) = ( (x₁+x₂)/2 , (y₁+y₂)/2 )
📐 4. Useful Checks
- Collinear points: three points are collinear if the sum of the two shorter distances equals the longest (they lie on one line).
- Type of triangle: compare side lengths — equilateral (all equal), isosceles (two equal), scalene (all different); right-angled if a² + b² = c².
🧠 Remember: Distance = √[(x₂−x₁)²+(y₂−y₁)²]. From origin = √(x²+y²). Section (m:n) = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)). Mid-point uses ratio 1:1. Trisection → ratios 1:2 and 2:1.
⭐ Exam Tips: ① Distance between two points. ② Check collinearity / type of triangle or quadrilateral. ③ Section formula (divide in a ratio). ④ Mid-point. ⑤ Points of trisection. ⑥ Find the ratio in which a point divides a segment.