Complete Solutions and Summary of Circles – NCERT Class 10, Mathematics, Chapter 10 – Summary, Questions, Answers, Extra Questions
Comprehensive summary and explanation of Chapter 10 'Circles', covering concepts such as definition and properties of circles, chords, arcs, sectors and segments, tangents and their properties, number of tangents from a point, lengths of tangents, relation between chords and radii, and proofs of important theorems along with multiple solved examples and exercises as per NCERT Class X Mathematics.
Updated: 8 months ago
Circles
Chapter 10: Mathematics - Ultimate Study Guide | NCERT Class 10 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Circles Class 10 NCERT
Overview & Key Concepts
- Chapter Goal: Understand circle properties, tangents, secants, and theorems on tangents. Exam Focus: Proofs, tangent lengths, applications. 2025 Updates: Real-life examples like pulleys, wheels. Fun Fact: 'Tangent' from Latin 'tangere' meaning to touch. Core Idea: Tangents perpendicular to radius, equal lengths from external point. Real-World: Bicycle wheels, pulleys for water drawing.
- Wider Scope: Geometry foundation, engineering applications in design and mechanics.
10.1 Introduction
- Recall from Class IX: Circle as points at constant distance (radius) from center. Terms: chord, segment, sector, arc.
- Line and circle interactions: Non-intersecting (no common points), secant (two common points), tangent (one common point).
- Examples: Fig. 10.1 shows three cases. Non-intersecting outside, secant crossing, tangent touching at one point.
- Real-life: Pulley rope as tangents (Fig. 10.2). Wheel on ground tangent to surface (Fig. 10.4).
- Study: Tangent existence and properties.
10.2 Tangent to a Circle
- Tangent intersects circle at exactly one point (point of contact).
- Activity 1: Rotate wire AB about P on circular wire. Positions intersect at two points except one tangent position A'B'. Shows one tangent per point; tangent as limit of secant when chord endpoints coincide.
- Activity 2: Draw secant PQ, parallels on both sides. Chords shorten to zero, forming two tangents parallel to PQ. Confirms at most two parallel tangents.
- Point of contact: Common point of tangent and circle.
- Theorem 10.1: Tangent perpendicular to radius at contact. Proof: OQ > OP for Q ≠ P on tangent XY, OP shortest distance, hence perpendicular (Theorem A1.7).
- Remarks: One tangent per point; radius line called normal.
- Exercise 10.1: Questions on tangents count, blanks, length calculation, drawing parallels.
10.3 Number of Tangents from a Point on a Circle
- Activity 3: Point inside: All lines intersect two points, no tangent. On circle: One tangent. Outside: Two tangents.
- Cases: Inside none, on one, outside two.
- Length of tangent: From external point to contact.
- Theorem 10.2: Equal tangent lengths from external point. Proof: Triangles OQP, ORP congruent (RHS), PQ=PR (CPCT). Alternative: Pythagoras PQ²=PR².
- Remarks: OP bisects angle QPR.
- Example 1: Chord AB of larger circle touches smaller concentric at P, bisected. Proof: OP ⊥ AB, bisects chord.
- Example 2: Tangents TP, TQ from T, prove ∠PTQ=2∠OPQ. Proof: TP=TQ isosceles, angles (180°-θ)/2=90°-θ/2, ∠OPQ=90°-(90°-θ/2)=θ/2.
- Example 3: Chord PQ=8 cm, radius 5 cm, tangents intersect at T, TP=? Proof: OR=3 cm, similar triangles, TP=20/3 cm.
- Exercise 10.2: MCQs on length, angles; proofs on parallels, perpendicular; radius from distance; concentric chords; circumscribed quadrilateral; parallel tangents angle; supplementary angles; rhombus; triangle sides; opposite sides supplementary.
10.4 Summary
- Tangent meaning.
- Perpendicular to radius.
- Equal lengths from external 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
- Tangents: Properties, proofs.
- Theorems: 10.1 perpendicular, 10.2 equal lengths.
- Applications: Real-life, geometry problems.
- Tip: Practice proofs; use activities for understanding.
Exam Case Studies
Proofs on tangents, lengths calculations, quadrilaterals circumscribing circles.
Project & Group Ideas
- Model tangents with wires; derive theorems.
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