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.

Key Themes & Tips

• Coordinates: Plotting, distances.
• Formulas: Distance, section.
• Applications: Collinear, shapes.
• Tip: Use positive roots; verify ratios.

Project & Group Ideas

• Plot maps; derive formulas.

Key Definitions & Terms - Complete Glossary

All terms from chapter; easy to memorize. Detailed explanations with examples for better understanding.

Tip: Use flashcards; visualize on graph paper.

Extended: Coordinate geometry bridges algebra-geometry; used in vectors, transformations. Historical: Rene Descartes invented in 1637.

Common Errors: Sign in differences; ratio order m:n vs n:m.

60+ Questions & Answers - NCERT Based (Class 10)

Structured as Part A (1 mark, short answers), Part B (4 marks, ~6 lines answers), Part C (8 marks, detailed). 20 per part, based on chapter content, with answers matching the mark scheme. Includes variations for depth.

Part A: 1 Mark Questions (Short Answers)

Part B: 4 Marks Questions (Answers in ~6 Lines)

Part C: 8 Marks Questions (Detailed Answers)

Practice Tip: Plot graphs; use formula sheet.

Important Formulas

Key equations for quick reference, with derivations and examples.

Use in: Problems for calculations. Derive from Pythagoras for distance, similarity for section.

Extended: For external division (higher), but internal focus here. Common: Ratio order affects coordinates.

Variations: 3D distance √(x²+y²+z²).

Theorems & Proofs - Detailed Explanations

Key theorems with step-by-step proofs and examples. Focus on foundational ones.

Tip: Visualize proofs on graph.

Extended: Apollonius theorem for medians, but chapter focus on basics. Errors: Assuming order in ratios.

Applications: Prove shapes using distances equal or angles 90° via Pythagoras.

Derivations - Step by Step

• Distance Formula: Step 1: Points P,Q. Step 2: Difference dx=x2-x1, dy=y2-y1. Step 3: Pythagoras PQ=√(dx²+dy²). Example: Derives from right triangle.
• Section Formula: Step 1: Ratio m:n. Step 2: Similarity PA/BP=m/n. Step 3: AQ/PC=m/n, etc. Step 4: Solve x=(m x2 + n x1)/(m+n). Similarly y.
• Mid-Point: Special m=n=1. Average coordinates.

Tip: Practice derivations for exams.

Extended: For external, sign change. Use in 3D: Add z term.

Common: Vector method alternative for section.

All Textbook Examples - Step by Step Solutions

Complete list with detailed explanations like the book, step by step.

Tip: Verify with graphs.

Extended: Variations like external division not in chapter, but note for higher.

All exercises hinted in summary.

Methods of Solving - Step by Step

• Distance: Plug differences, square root. Handle signs via squares.
• Section: Identify ratio, plug formula. For mid, average.
• Collinear: Distances or area=0 (1/2[x1(y2-y3)+...]=0).
• Equidistant: Set equal, solve equation.

Choose: Formula direct; graph for visual.

Extended: Software like GeoGebra. Errors: Ratio mix-up.

Applications: In proofs for shapes.

Applications & Word Problems - Real Life

• Distances: Maps, towns. Example: Town A to B.
• Divisions: Tower positions, mid-points. Example: Telephone tower.
• Shapes: Verify triangles, squares. Example: Desks collinear.
• Equidistant: Perpendicular bisectors. Example: y-axis point.
• Navigation: Coordinates in GPS.
• Art/Engineering: Plot designs.

Tip: Draw diagrams; positive values.

Extended: In physics for trajectories, seismology for epicenters.

Problems: Classroom flags, parallelogram vertices.

Quick Revision Notes & Mnemonics - Exam-Ready

Mnemonics: Distance: Pythagoras Always (PA for points A). Section: More Near (m x2 near B).

Tip: Graph paper; check calculations.