NCERT Solutions for Class 10 Maths Chapter 10: Circles
Circles have fascinated mathematicians for thousands of years, and Chapter 10 of Class 10 NCERT Maths gets straight to the most important relationships involving circles and straight lines. While students in earlier classes learned about basic circle properties like radius, diameter, chord, and arc, this chapter takes a sharp turn toward something new — the concept of a tangent. A tangent is a line that touches a circle at exactly one point, and this single idea opens up a rich set of theorems and problem-solving techniques. The chapter focuses on two major theorems. For other chapters of NCERT Solutions for Class 10 Maths, check NCERT Solutions for Class 10 and NCERT Solutions.
The first tells us that the tangent to a circle at any point is perpendicular to the radius drawn to that point. The second tells us something beautiful — if two tangents are drawn from an external point to a circle, both tangents are equal in length. These theorems are proved in the NCERT textbook, and understanding the proofs — not just the results — is what distinguishes students who score full marks from those who do not. Chapter 10 has two exercises with a total of 17 questions, and the chapter is known for being conceptually clean and approachable. Myclass24 offers detailed NCERT Solutions for Class 10 Maths Chapter 10 Circles with theorem proofs, diagrams, and step-by-step solutions that align with CBSE board marking standards.
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Chapter 10 Circles – Theorems, Tangent Properties & Exercise Details
This chapter is built around the relationship between a circle and tangents drawn to it. To answer questions from this chapter confidently, students need a clear understanding of what a tangent is, how it behaves relative to the radius, and how the "equal tangents from external point" property is used in problems.
Key Terminology in Chapter 10
| Term | Definition |
|---|---|
| Tangent | A line that touches the circle at exactly one point (called the point of tangency) |
| Secant | A line that intersects the circle at two distinct points |
| Point of Contact | The single point where a tangent meets the circle |
| External Point | A point outside the circle from which tangents can be drawn |
| Length of Tangent | Distance from the external point to the point of contact |
The Two Main Theorems of Chapter 10
| Theorem | Statement | Key Proof Method |
|---|---|---|
| Theorem 10.1 | The tangent at any point of a circle is perpendicular to the radius through the point of contact | Proof by contradiction — assume tangent is not ⊥ to radius, show this leads to a contradiction |
| Theorem 10.2 | The lengths of tangents drawn from an external point to a circle are equal | Using congruence of right triangles formed by radius, tangent, and line joining centre to external point |
Number of Tangents Possible from a Point
| Position of Point | Number of Tangents | Reason |
|---|---|---|
| Inside the circle | 0 (Zero) | Any line through an interior point cuts the circle at two points — no tangent possible |
| On the circle | 1 (One) | Exactly one tangent at the point of contact |
| Outside the circle | 2 (Two) | Two equal-length tangents can always be drawn from an external point |
Exercise-wise Breakdown of Chapter 10
| Exercise | Focus Area | No. of Questions | Types of Problems |
|---|---|---|---|
| Exercise 10.1 | Basic tangent concepts, number of tangents | 4 | MCQ-type, conceptual, short proofs |
| Exercise 10.2 | Theorem application, tangent lengths, mixed geometry | 13 | Proof-based, calculation, quadrilateral/triangle involving tangents |
Important Results Used in Exercise 10.2
- If PA and PB are two tangents from external point P, then PA = PB (Theorem 10.2).
- OA ⊥ PA and OB ⊥ PB (radius is perpendicular to tangent) — so ∠OAP = ∠OBP = 90°.
- Quadrilateral OAPB (where O is centre, P is external point) has angle sum = 360°. Since ∠OAP + ∠OBP = 180°, the remaining two angles ∠AOB + ∠APB = 180°.
- When a circle is inscribed in a triangle or quadrilateral, the tangent lengths from each vertex are equal — this is used heavily in Exercise 10.2 Questions 10 to 13.
- For a quadrilateral circumscribed about a circle: AB + CD = BC + DA (sum of opposite sides are equal).
Chapter 10 is one of those chapters where careful diagram-drawing makes all the difference. Many students lose marks not because they do not know the theorem, but because they cannot translate the word problem into a correctly drawn circle with labelled tangents, radii, and angles. Myclass24's NCERT Solutions for Class 10 Maths Chapter 10 Circles include properly constructed geometric figures with every solution, helping students develop the diagram habit that CBSE examiners reward.
NCERT Solutions for Class 10 Maths Chapter 10: Circles
The Circles chapter focuses on tangents to circles and their important properties. Students learn how tangents interact with circles and how geometric relationships can be used to solve problems. The chapter explains that a tangent touches a circle at exactly one point and introduces several theorems related to tangents. Understanding these concepts helps students develop logical reasoning and geometric problem-solving skills. This chapter is important because circles are widely used in geometry and have practical applications in engineering, design, and architecture. Many board examination questions are based on tangent properties and related proofs.
A tangent to a circle is a straight line that touches the circle at exactly one point, known as the point of contact. Unlike a secant, which intersects the circle at two points, a tangent only touches it once. Tangents are important because they help establish various geometric relationships used in proofs and calculations. Students learn how tangents behave and how they can be identified in geometric figures. Understanding tangents forms the foundation for solving most questions in this chapter and is frequently tested in examinations.
One of the most important theorems in this chapter states that the radius drawn to the point of contact of a tangent is perpendicular to the tangent. This property helps solve many geometry problems involving circles and tangents. Students use this theorem to establish right angles and calculate unknown lengths. Understanding why this relationship exists strengthens geometric reasoning and proof-writing skills. Examination questions often require students to apply this theorem directly or use it as part of a larger geometric proof.
Yes, two tangents can be drawn from a point outside a circle. These tangents touch the circle at different points and have special properties. One important property is that the lengths of the tangents drawn from the same external point are equal. This theorem is frequently used to solve geometry problems involving unknown lengths and angles. Understanding this concept helps students recognize patterns in circle-related figures and apply mathematical reasoning effectively. It is one of the most commonly tested concepts in board examinations.
When two tangents are drawn from the same external point, the line segments between the external point and their respective points of contact are equal in length. This property is proven using congruent triangles formed by the radii and tangent segments. The theorem simplifies many geometric calculations and helps establish relationships between different parts of a figure. Students frequently encounter questions requiring the use of this property. Mastering this theorem improves confidence and accuracy when solving circle-related geometry problems.
Circles are used extensively in engineering, architecture, transportation, and design. Wheels, gears, clocks, pipelines, and many mechanical components are based on circular shapes. Understanding the geometric properties of circles helps professionals design structures and machines efficiently. Learning about tangents also has applications in navigation, construction, and technical drawing. These real-life examples demonstrate that the concepts studied in this chapter are not limited to textbooks but are widely used in practical situations and technological developments.
Students often confuse tangents with secants or forget important tangent properties while solving questions. Another common mistake is failing to identify the correct point of contact. Errors also occur when applying the theorem regarding equal tangent lengths. To avoid these mistakes, students should carefully analyze diagrams and revise key theorems regularly. Drawing neat figures and labeling points correctly can improve understanding. Consistent practice helps students solve circle problems more accurately and perform better in examinations.