Mathematics · Class 10
Circles
Tangent properties, number of tangents from external point.
3 sections15–25 min read6 MCQs · 2 examples
Chapter Overview
Tangent ⊥ radius at point of contact — this right angle is the key to most tangent problems.
Class 10 Circles focuses on tangents — lines touching the circle at exactly one point. You study properties of tangents, angles between tangents, and the number of tangents from an external point.
Tangent Properties
- Tangent touches circle at exactly one point (point of contact)
- Tangent is perpendicular to radius at point of contact
- Tangents from external point are equal in length
- At most one tangent at any point on the circle
Key Results
- Two tangents can be drawn from external point
- No tangent from point inside circle
- Exactly one tangent from point on circle
- Angle between tangents and centre properties in quadrilateral formed
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Tangent length
From point P, 13 cm from centre O of circle of radius 5 cm, find tangent length PT.
- 1OT ⊥ PT, so ΔOTP is right-angled
- 2PT² = OP² − OT² = 169 − 25 = 144
- 3PT = 12 cm
Answer
12 cm
Example 2: Equal tangents
PA and PB are tangents from P to circle. If PA = 8 cm, find PB.
- 1Tangents from external point are equal
- 2PB = PA
Answer
PB = 8 cm
Key Points to Remember
- ✓ Radius ⊥ tangent at contact point
- ✓ Use Pythagoras in right triangle with radius and distance from centre
- ✓ Tangent segments from same external point are equal
Exam Tips
- • Mark right angle at point of contact
- • Draw both tangents from external point
- • Prove equal tangents using congruent triangles (RHS)
Practice MCQs — Circles
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Question 1 of 6Score: 0/0
Tangent to circle intersects at: