Coordinate Geometry — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Coordinate Geometry (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Coordinate Geometry (CBSE & HBSE)
Coordinate Geometry combines algebra and geometry. CBSE tests distance, section, and area formulas with application problems. HBSE focuses on direct formula application.
Distance and Section Formula
Distance Formula
Distance between P(x₁, y₁) and Q(x₂, y₂): $$PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
Derived from Pythagoras: Draw a right triangle with PQ as hypotenuse.
Special cases:
- Distance from origin O(0,0) to P(x,y): OP = √(x²+y²)
- Distance between (x₁,0) and (x₂,0) (on x-axis): |x₂-x₁|
Uses of Distance Formula
| Geometry Problem | Method |
|---|---|
| Prove isosceles triangle | Show two sides equal |
| Prove equilateral triangle | Show all three sides equal |
| Prove right triangle | Check if a²+b²=c² |
| Prove parallelogram | Show opposite sides equal |
| Prove rhombus | All sides equal |
| Prove square | All sides equal + diagonals equal |
Section Formula
Point P that divides segment joining A(x₁,y₁) and B(x₂,y₂) in ratio m:n:
Internally: $P = \left(\frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n}\right)$
Midpoint Formula (m=n=1): $M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
Key Applications
- Centroid of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃):
- Centroid divides each median in ratio 2:1
G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
CBSE Tip: The section formula requires identifying which point is A and which is B, and what ratio m:n is. Be careful with the order!
Area of Triangle and Collinearity
Area of a Triangle Using Coordinates
For vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃): $$\text{Area} = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$$
Memory Aid: The formula uses a 'zigzag' pattern across the vertices.
Signed form (without |...|) gives positive or negative area — the absolute value gives the actual area.
Collinearity of Three Points
Three points are collinear (lie on one straight line) if and only if: $$\text{Area of triangle} = 0$$ $$x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) = 0$$
Example: Are A(1,2), B(4,8), C(7,14) collinear? Area = 1/2|1(8-14) + 4(14-2) + 7(2-8)| = 1/2|-6 + 48 - 42| = 1/2|0| = 0 → Collinear! ✓
Using Area to Find Unknown Coordinate
If three points are collinear and one coordinate is unknown, set Area = 0 and solve.
Example: A(1,2), B(2,k), C(4,5) are collinear. Find k. 1(k-5) + 2(5-2) + 4(2-k) = 0 k - 5 + 6 + 8 - 4k = 0 -3k + 9 = 0 → k = 3
Area of Quadrilateral
Split into 2 triangles using one diagonal, find areas, and add.
HBSE Tip: When using the area formula, be consistent with the order of vertices (either all clockwise or all counterclockwise). The absolute value ensures a positive area.
Applications and Graph Interpretations
Using Distance Formula to Prove Geometric Shapes
Proving it is a Rectangle:
- Show all 4 sides are NOT equal (not a square)
- Show opposite sides are equal AND diagonals are equal
Proving it is a Square:
- Show all 4 sides are equal
- Show diagonals are equal (not just any rhombus)
Proving it is a Parallelogram:
- Show opposite sides are equal: AB = CD AND BC = DA
Proving it is a Rhombus:
- Show all 4 sides are equal
Proving it is NOT a specific shape:
- If all sides equal but diagonals unequal → Rhombus (not square)
Finding Points on Axes
| Point on x-axis | y-coordinate = 0 → point is (x, 0) |
|---|---|
| Point on y-axis | x-coordinate = 0 → point is (0, y) |
Example: Find point on x-axis equidistant from A(3,4) and B(-5,-2).
- Let P = (x, 0)
- PA = PB → (x-3)² + 16 = (x+5)² + 4
- x² - 6x + 25 = x² + 10x + 29
- -16x = 4 → x = -1/4
- P = (-1/4, 0)
Locus Problems
Locus: Set of all points satisfying a given condition.
- Locus equidistant from two points = perpendicular bisector of the segment joining them
- In coordinate geometry, find by setting distances equal
CBSE Application: Board questions often involve 'Find a point on y-axis equidistant from two given points' — this uses PA = PB with P = (0, y).
Frequently asked questions
Are these Coordinate Geometry notes free?
Yes — the Coordinate Geometry notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Coordinate Geometry notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Coordinate Geometry chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Coordinate Geometry.