Class 10 Maths Circles Complete Guide 2027 — Tangent Theorems, Proofs, Board Questions

In short: Circles carries 4 marks in CBSE Class 10 Maths 2027 under the Geometry unit. The chapter focuses entirely on tangents — two theorems (tangent is perpendicular to radius, tangent lengths from an external point are equal), their proofs, and numerical problems. It is proof-heavy, so understanding the logic matters more than memorising formulas. Master the two theorems and their applications, and these are some of the most predictable marks on the paper.

Circles is Chapter 10 in CBSE Class 10 Maths. It falls under the Geometry unit (15 marks total) and specifically carries about 4 marks. Unlike Class 9 Circles, which dealt with chords and arcs, Class 10 is entirely about tangents and their properties. There are only two main theorems, but board exams test them through proofs, constructions, and numerical problems.

This is a theorem-driven chapter. If you understand WHY the theorems work, the proofs and problems become straightforward. For the full marks distribution, see the CBSE Class 10 chapter-wise weightage.

Key Definitions

• Tangent: A line that touches the circle at exactly one point. This point is called the point of tangency or point of contact.
• Secant: A line that intersects the circle at two points.
• Normal: The line perpendicular to the tangent at the point of contact. The normal always passes through the centre.

Important fact: At any point on a circle, there is exactly one tangent. From a point outside the circle, exactly two tangents can be drawn.

The Two Theorems

Theorem 1: Tangent is Perpendicular to Radius

Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

If OA is the radius and PA is the tangent at point A, then OA ⊥ PA (angle OAP = 90°).

Proof approach (for board exams):

1. Let O be the centre and P the point of contact
2. Assume the tangent is the line l at point P
3. Take any other point Q on line l
4. Since Q is on the tangent (not on the circle), OQ > OP (Q is outside the circle, P is on it)
5. This is true for every point Q on l other than P
6. So OP is the shortest distance from O to line l
7. The shortest distance from a point to a line is the perpendicular
8. Therefore OP ⊥ l

Why this matters: This 90° angle is the foundation for almost every numerical problem in the chapter. Whenever you see a tangent meeting a radius, you have a right angle — and that means Pythagoras theorem applies.

Theorem 2: Equal Tangent Lengths from External Point

Statement: The lengths of tangents drawn from an external point to a circle are equal.

If PA and PB are tangents from external point P to a circle with centre O, touching the circle at A and B respectively, then PA = PB.

Proof (frequently asked for 3-4 marks):

1. In triangles OAP and OBP:
2. OA = OB (radii of the same circle)
3. OP = OP (common side)
4. Angle OAP = Angle OBP = 90° (Theorem 1: tangent ⊥ radius)
5. By RHS congruence: △OAP ≅ △OBP
6. Therefore PA = PB (corresponding parts of congruent triangles)

Bonus from this proof: Angle OPA = Angle OPB. This means OP bisects the angle between the two tangents (angle APB). This property appears in problems.

Board Exam Question Types

Type 1: Prove the Theorem (3-4 marks)

Either theorem can be asked as a direct proof. Write it step-by-step with clear reasoning. Board examiners give step-wise marks, so show every line.

Type 2: Find Tangent Length Using Pythagoras (2-3 marks)

"A tangent PA is drawn from point P to a circle with centre O and radius 5 cm. If OP = 13 cm, find PA."

Since tangent ⊥ radius: angle OAP = 90° By Pythagoras: PA² = OP² − OA² = 169 − 25 = 144 PA = 12 cm

This is the most common numerical type. Always form the right triangle with the radius, tangent, and the line from centre to external point.

Type 3: Tangent Properties in Quadrilateral/Triangle (3-4 marks)

"Prove that the tangents drawn at the ends of a diameter are parallel."

Let AB be a diameter. Tangent at A meets a line at P, tangent at B meets the line at Q. OA ⊥ PA and OB ⊥ PB (Theorem 1). Since OA and OB are parts of the same line (diameter), PA and QB are both perpendicular to the same line. Two lines perpendicular to the same line are parallel. Therefore PA ∥ QB.

Type 4: Circle Inscribed in a Triangle (3-5 marks)

"Prove that in a triangle, the tangent lengths from each vertex satisfy: AB + CD = AC + BD where the incircle touches sides at specific points."

This type uses Theorem 2 repeatedly. Label the points where the incircle touches each side, then use equal tangent lengths from each vertex to set up equations.

Common Mistakes That Cost Marks

1. Forgetting to use the 90° angle. Every tangent-radius junction gives you a right angle. If you are stuck on a numerical problem, draw the radius to the point of tangency — you now have a right triangle and Pythagoras applies.

2. Incomplete proofs. Writing "PA = PB because tangent lengths are equal" without the proof steps earns zero marks if the question says "Prove." You must show OA = OB (radii), OP = OP (common), angle = 90° (Theorem 1), then RHS congruence, then CPCT.

3. Not stating the theorem used. Board examiners expect you to name the theorem. Write "By Theorem 10.1, the tangent is perpendicular to the radius" — not just "angle = 90°."

4. Confusing tangent with secant in diagrams. A tangent touches at one point. A secant cuts through at two points. If your diagram shows the line crossing the circle, it is a secant, not a tangent.

5. Not labelling diagrams clearly. Mark the centre O, the point of tangency, the external point, and the right angle symbol (□). Unlabelled diagrams make it hard for the examiner to follow your proof.

4-Week Practice Plan

Week 1: Definitions and Theorem 1

• Read NCERT Section 10.1 and 10.2
• Understand Theorem 1 proof — write it from memory 3 times during the week
• Solve 10 basic problems on finding tangent length using Pythagoras
• Draw diagrams for every problem — make this a habit

Week 2: Theorem 2 and Proofs

• Master Theorem 2 proof — write it from memory until fluent
• Solve NCERT Exercise 10.1 and 10.2 completely
• Practice the "tangents from external point" type problems
• Attempt 5 "prove that" questions from sample papers

Week 3: Applications — Inscribed Circles and Constructions

• Solve problems involving circles inscribed in triangles and quadrilaterals
• Practice using equal tangent lengths to set up simultaneous equations
• Link to Constructions for tangent construction problems
• Attempt CBSE important questions for this chapter

Week 4: Board Paper Practice

• Solve Circles questions from the last 5 years of CBSE board papers
• Focus on 3-4 mark proof questions — these are the high-value targets
• Time yourself: proofs should take 5-6 minutes maximum
• Final drill: write both theorem proofs from memory without any reference

Quick Revision Card for Exam Day

• Theorem 1: Tangent ⊥ Radius at point of contact (angle = 90°)
• Theorem 2: Tangent lengths from external point are equal (PA = PB)
• Proof method for Theorem 2: RHS congruence (OA=OB radii, OP=OP common, angles 90°)
• OP bisects angle APB (bonus from Theorem 2 proof)
• Tangent + Radius = Right triangle → use Pythagoras
• From external point: exactly 2 tangents. On circle: exactly 1 tangent
• Tangents at ends of a diameter are parallel
• Always draw and label your diagram with the right angle symbol
• Name the theorem in your answer

For guided practice on circle theorems and tangent proofs with step-by-step hints, try Padhaao's AI tutor — it checks each step of your proof and points out missing reasoning.

Frequently Asked Questions

How many marks does Circles carry in CBSE Class 10 Maths 2027?

Circles carries approximately 4 marks in the CBSE Class 10 board exam 2027, as part of the Geometry unit (15 marks total). You can expect one 3-4 mark question, usually asking to prove one of the two tangent theorems or to solve a numerical problem involving tangent lengths and the Pythagoras theorem.

What are the two main theorems in CBSE Class 10 Circles?

The two theorems are: (1) The tangent at any point of a circle is perpendicular to the radius through the point of contact, and (2) The lengths of tangents drawn from an external point to a circle are equal. Every problem in this chapter is built on one or both of these theorems.

How do I prove that tangent lengths from an external point are equal?

Use RHS (Right angle-Hypotenuse-Side) congruence. In triangles OAP and OBP, show that OA = OB (both radii), OP = OP (common side), and angle OAP = angle OBP = 90° (tangent perpendicular to radius). By RHS, the triangles are congruent, so PA = PB by CPCT (Corresponding Parts of Congruent Triangles).

Is NCERT enough for Circles in CBSE Class 10?

Yes, NCERT is sufficient. The chapter has only 2 exercises with about 12 questions total. Solve all of them thoroughly, then practise 3-5 years of previous year board questions. The board exam questions closely follow NCERT patterns — especially theorem proofs and Pythagoras-based numerical problems.

Why is the 90° angle between tangent and radius so important?

Because it creates a right triangle in every tangent problem. Whenever you have a tangent, radius, and a line from the centre to an external point, you have a right triangle where you can apply the Pythagoras theorem. This 90° angle is used in almost every numerical problem in the chapter and is also the foundation for proving Theorem 2.

Circles is a compact chapter with just two theorems, but the proofs and applications test your ability to think logically. Know the theorems, write clean proofs with clear reasoning, and always draw labelled diagrams with the right-angle symbol.