Class 9 Maths Chapter 10 Introduction to Circles NCERT Solutions: All Exercises Tangents Chords Step-by-Step

Introduction

Chapter 10 finally brings circles into your geometry toolkit. You'll learn key theorems about tangents (touching at one point), chords (line segments inside), and angles. The big ones are tangents from same external point equal length, chords equidistant from center equal, and angle in semicircle 90 degrees.

Two exercises with proofs and applications. Draw circles accurately - compass work matters here.

Key Theorems and Properties

Tangent perpendicular to radius at point of contact
Tangents from external point equal: PT = QT (P,Q tangents from T)
Chords equidistant from center equal length
Equal chords subtend equal angles at center
Angle in semicircle = 90°
Perpendicular from center bisects chord

Exercise 10.1 Solved Questions

Question 1: How many tangents from external point? Two.

Solution: Two tangents touch at different points.

Question 2: Tangents PA, PB from P to circle center O. Prove PA = PB.

Solution: OP perpendicular PA,PB (tangent perp radius). Triangles OPA=OPB (RHS: OP common, right angles, OA=OB radii). PA=PB (CPCT).

Question 3: From P, two tangents touch at A,B. O center. Prove angle OAP = angle OBP.

Solution: Same triangles OPA=OPB → equal angles.

Question 4: Tangents from different points equal? No.

Solution: Different external points, different tangent lengths.

Question 5: Number of tangents from center? None (infinite points).

Question 6: Circle radius 4cm. Tangent length from 5cm away?

Solution: Pythagoras: sqrt(5²-4²)=sqrt(25-16)=3cm.

Exercise 10.2 Solved Questions

Question 1: Chords AB=CD. Prove equal distance from center.

Solution: Perp from O bisects chords: OM perp AB, ON perp CD at midpoints. Triangles OMB=ONC (OM=ON radii, MB=NC half chords equal) → perpendicular distances equal.

Question 2: Equal chords subtend equal angles at center.

Solution: Same triangles isosceles → base angles equal.

Question 3: Perp from center bisects chord. Prove.

Solution: Isosceles triangles OAM=OBM → AM=BM.

Question 4: Angle in semicircle 90°. Prove.

Solution: Angle at circumference = half angle at center (same arc). Semicircle arc 180° → angle 90°.

Question 5: Equal chords equal arcs.

Solution: Equal central angles → equal arcs.

Question 6: Fig 10.20: Prove angle in same segment equal.

Solution: Angles subtended same arc equal.

Extra Practice Questions

Extra Q1: Circle radius 5cm, external point 13cm away. Tangent length? 12cm.

Extra Q2: Chord 8cm, distance from center 6cm. Radius? 10cm.

Extra Q3: Two chords 10cm each, angles at center 60°, 120°. Same? No.

Extra Q4: Semicircle diameter AB. Angle ACB? 90° (C on circumference).

Extra Q5: Tangents from P touch at A,B. OA perp PA. Find angle OAP if OP=13, OA=5. sin^-1(5/13).

15 FAQs

Tangent touches circle? One point.
Tangents from external point? Equal length.
Radius to tangent? Perpendicular.
Equal chords from center? Equal distance.
Perp from center to chord? Bisects chord.
Angle in semicircle? 90 degrees.
Equal chords subtend? Equal central angles.
Tangent length formula? sqrt(power)=sqrt(OP^2-r^2).
Number tangents from center? None.
Arc equal when? Chords equal.
Angle same segment? Equal.
Chord bisector property? Perp from center.
Two tangents different points? Different lengths.
Inscribed angle theorem? Half central angle.
Circle center to chord shortest? Perpendicular distance.

Circles unlock lots of geometry ahead. Check www.fuzymathacademy.com for my compass drawing videos and live tangent problems. You'll master Chapter 10 fast.