Complete Solutions and Summary of Coordinate Geometry – NCERT Class 10, Mathematics, Chapter 7 – Summary, Questions, Answers, Extra Questions
Comprehensive summary and explanation of Chapter 7 'Coordinate Geometry', covering concepts of the Cartesian plane, distance formula, section formula, mid-point formula, coordinate-based applications like collinearity, area, and trisection, with multiple solved examples and extra practice questions aligned with NCERT Class X Mathematics.
Updated: 8 months ago
Coordinate Geometry
Chapter 7: Mathematics - Ultimate Study Guide | NCERT Class 10 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Coordinate Geometry Class 10 NCERT
Overview & Key Concepts
- Chapter Goal: Learn to find distances between points and divide line segments in ratios using coordinates. Exam Focus: Distance formula, section formula, applications. 2025 Updates: Emphasis on real-life plotting. Fun Fact: Descartes linked algebra and geometry. Core Idea: Coordinates represent positions algebraically. Real-World: Town distances, classroom seating.
- Wider Scope: Foundation for higher math, physics vectors.
7.1 Introduction
- Recall Class IX: Point position needs axes. x-coordinate (abscissa) from y-axis, y-coordinate (ordinate) from x-axis. Points on x-axis (x,0), y-axis (0,y).
- Activity: Plot A(4,8), B(3,9), C(3,8), D(1,6), E(1,5), F(3,3), G(6,3), H(8,5), I(8,6), J(6,8), K(6,9), L(5,8) and join to A. Then P(3.5,7), Q(3,6), R(4,6) triangle; X(5.5,7), Y(5,6), Z(6,6) triangle; S(4,5), T(4.5,4), U(5,5) triangle; S to (0,5),(0,6); U to (9,5),(9,6). Forms a picture (e.g., boat).
- Linear equations graph lines; quadratics parabolas. Coordinate geometry studies figures algebraically, applied in physics, engineering, navigation, seismology, art.
- Chapter covers: Distance between points, point dividing segment in ratio.
7.2 Distance Formula
- Situation: Town B 36 km east, 15 km north of A. Distance using Pythagoras: √(36² + 15²) = √(1296 + 225) = √1521 = 39 km.
- Points on x-axis: A(4,0), B(6,0); AB=6-4=2 units.
- On y-axis: C(0,3), D(0,8); CD=8-3=5 units.
- Between axes: A(4,0) to C(0,3); AC=√(4² + 3²)=5 units.
- General: P(4,6), Q(6,8); Drop perpendiculars, use Pythagoras: PQ=√[(6-4)² + (8-6)²]=√8=2√2.
- Different quadrants: P(6,4), Q(-5,-3); PT=11, QT=7; PQ=√(11² + 7²)=√170.
- Formula derivation: P(x1,y1), Q(x2,y2); PT=|x2-x1|, QT=|y2-y1|; PQ=√[(x2-x1)² + (y2-y1)²].
- From origin: OP=√(x² + y²).
Example 1: Points (3,2), (-2,-3), (2,3) Form Triangle?
- PQ≈7.07, QR≈7.21, PR≈1.41. Sum any two > third, yes triangle. PQ² + PR² = QR², right-angled at P.
Example 2: Points (1,7), (4,2), (-1,-1), (-4,4) Square?
- Sides AB=BC=CD=DA=√34, diagonals AC=BD=√68. All equal, diagonals equal, square. Alternative: AD² + DC² = AC², angle 90°.
Example 3: Desks A(3,1), B(6,4), C(8,6) Collinear?
- AB=3√2, BC=2√2, AC=5√2. AB + BC = AC, collinear.
Example 4: Relation x,y Equidistant (7,1),(3,5)
- (x-7)² + (y-1)² = (x-3)² + (y-5)² → x - y =2.
Example 5: Point on y-axis Equidistant (6,5),(-4,3)
- (0,y); 36 + (5-y)² =16 + (3-y)² → y=9. Point (0,9).
Exercise 7.1
- Distances, collinear checks, triangle types, square checks, equidistant points.
7.3 Section Formula
- Situation: Tower P divides AB (A(0,0), B(36,15)) in 1:2. Coordinates (12,5).
- Derivation: P divides A(x1,y1), B(x2,y2) in m1:m2. Using similarity: x=(m1 x2 + m2 x1)/(m1+m2), y=(m1 y2 + m2 y1)/(m1+m2).
- Mid-point: Ratio 1:1, ( (x1+x2)/2 , (y1+y2)/2 ).
Example 6: Divide (4,-3),(8,5) in 3:1
- x=(3*8 +1*4)/4=7, y=(3*5 +1*(-3))/4=3. Point (7,3).
Example 7: Ratio (-4,6) Divides (-6,10),(3,-8)
- m1:m2=2:7.
Example 8: Trisection (2,-2),(-7,4)
- 1:2 → (-1,0); 2:1 → (-4,2).
Example 9: y-axis Divides (5,-6),(-1,-4)
- Ratio 5:1, point (0,-13/3).
Example 10: Parallelogram (6,1),(8,2),(9,4),(p,3)
- Mid-point AC=BD → p=7.
Exercise 7.2
- Divisions, trisections, ratios, parallelograms, mid-points, areas.
7.4 Summary
- Distance, section formulas, mid-point.
Why This Guide Stands Out
Complete chapter coverage: Notes, examples, Q&A (all NCERT + extras), quiz. Student-centric, exam-ready for 2025. Free & ad-free.
Key Themes & Tips
- Coordinates: Plotting, distances.
- Formulas: Distance, section.
- Applications: Collinear, shapes.
- Tip: Use positive roots; verify ratios.
Exam Case Studies
Word problems on distances, divisions; shape verifications.
Project & Group Ideas
- Plot maps; derive formulas.
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