Chapter 7 — Coordinate Geometry

Chapter 7 of the Class 10 Maths NCERT textbook, "Coordinate Geometry", teaches students to find the distance between two points using the Distance Formula and the coordinates of a point dividing a line segment in a given ratio using the Section Formula.

• Geometry through algebra — Coordinate geometry places shapes on a plane with numbered axes so that lengths and positions can be handled with algebra. This bridge lets geometric questions be answered by calculation.
• The Distance Formula — Built from the Pythagoras theorem, the Distance Formula measures the straight-line gap between any two points from their coordinates, and its special case gives the distance of a point from the origin.
• Dividing a segment — The Section Formula pinpoints where a point falls when it splits a segment in a chosen ratio. Setting that ratio to 1:1 recovers the midpoint, the exact centre of the segment.
• Testing collinearity — Distances also reveal whether three points lie on one line: if the two shorter distances add up to the longest, the points are collinear. This supports classifying triangles and quadrilaterals.

1. 01Distance Formula: the distance between P(x₁,y₁) and Q(x₂,y₂) is √[(x₂–x₁)²+(y₂–y₁)²], derived from the Pythagoras theorem.
2. 02Distance of a point P(x,y) from the origin O(0,0) is √(x²+y²).
3. 03Section Formula: the point dividing segment AB — where A is (x₁,y₁) and B is (x₂,y₂) — internally in ratio m₁:m₂ has coordinates ((m₁x₂+m₂x₁)/(m₁+m₂), (m₁y₂+m₂y₁)/(m₁+m₂)).
4. 04Midpoint Formula (special case of Section Formula with ratio 1:1): midpoint of AB is ((x₁+x₂)/2, (y₁+y₂)/2).
5. 05Three points are collinear if the sum of the distances between the two pairs of adjacent points equals the distance between the outermost pair.
6. 06Coordinate geometry links algebra and geometry, with applications in physics, engineering, navigation, seismology, and art.