Textbook chapter 10
Understand tangency and use the two tangent theorems to find unknown lengths and angles.
Textbook exercise solutions
- Exercise 10.1 — 4 question groups
- Exercise 10.2 — 13 question groups
Videos, notes and tests
English · Session 2026–27 · NCERT-aligned study guide
Chapter overview
Understand tangency and use the two tangent theorems to find unknown lengths and angles.
Before you begin: Circle radius and chord, perpendicular distance, congruence and Pythagoras.
Key concepts · Solved examples · Practice · Mind map · Timed chapter test
Learn the key concepts
Tangent and secant
A tangent touches a circle at one point; a secant cuts it at two. From an exterior point there are two tangents, from a point on the circle one, and from an interior point none.
Why a radius is perpendicular
Let a tangent touch at T with centre O. Drop the perpendicular from O to the tangent line. If its foot were a point other than T, the perpendicular distance would be less than radius OT. The line would then pass through the circle and intersect twice, contradicting tangency. Therefore the perpendicular foot is T.
Why two tangent lengths are equal
For exterior P with contact points A,B, join OA,OB and OP. Triangles OAP,OBP are right triangles with common hypotenuse OP and equal radii OA=OB. They are congruent by RHS, so PA=PB. The same congruence also shows OP bisects the angle between the tangents.
Formula and result checklist
- OT ⟂ PT at contact T.
- From exterior P: PA=PB.
- PT²=OP²−r².
Solved examples
Example 1
A circle has radius 5 cm. An external point is 13 cm from its centre. Find the tangent length.
The radius is perpendicular to the tangent. PT²=13²−5²=144, so PT=12 cm.
Example 2
Tangents PA and PB have lengths 3x+1 and 5x−7. Find x and the lengths.
Equal tangents give 3x+1=5x−7. Thus x=4 and both lengths are 13 units.
Example 3
Two tangents PA,PB make 70° at P. Find central angle AOB.
In quadrilateral OAPB, angles at A and B are 90°. Hence angle AOB=360°−90°−90°−70°=110°.
Try it yourself
Work on paper before opening each answer. Show the reasoning and include units when needed.
1. Radius=8, centre-to-point distance=17. Find tangent length.
Check answer 1
15 units.
2. How many tangents can be drawn from a point inside a circle?
Check answer 2
None.
3. PA=2x+3, PB=4x−5. Find x.
Check answer 3
x=4; both tangent lengths are 11.
Common mistakes to avoid
- A radius is perpendicular only at the point of contact with that tangent.
- Equal tangents must be drawn from the same external point.
Chapter mind map
View this chapter in the Mind Maps library · Open full-size image
Check your understanding
Take the 6-question, 10-minute chapter practice test. Each correct answer earns 1 mark; there is no negative marking. See explanations after submission.
Start chapter 10 practice test →
Continue learning
Read the corresponding chapter in the official NCERT textbook, then complete its exercises on paper. These NSL explanations and original practice questions support your textbook study; they are not an exercise-by-exercise solution manual.
Official CBSE syllabus 2026–27 · All chapters · Previous chapter · Next chapter
Check what you have learned
Finished the topics? Practise with the free chapter test.
Start free test