CBSE Class 10 Mathematics Chapter 10: Circles NCERT Solutions

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NCERT Solutions PDF Class 10 PDF

This chapter delves into the fundamental properties of circles in Class 10 Mathematics, as per the CBSE curriculum. The NCERT Solutions for Chapter 10 (Circles) provide clear explanations and step-by-step solutions for exercises. Key concepts covered include understanding tangents to a circle, the relationship between the radius and tangent at the point of contact, and distinguishing between tangents and secants. The solutions also address how to calculate the length of a tangent segment using the Pythagorean theorem. These resources are designed to help students grasp the geometric principles related to circles, build problem-solving skills, and prepare effectively for their board examinations by offering a comprehensive review of the chapter's content.

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Quick info

BoardCBSE
ClassClass 10
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 25

Chapter summary

Chapter 10, Circles, for Class 10 Mathematics focuses on the properties of tangents and secants. The NCERT Solutions cover definitions, theorems related to tangents, and practical applications. Students will learn to identify tangents, secants, and points of contact, and solve problems involving lengths and parallel lines related to a circle. This chapter is crucial for understanding the geometry of circles.

Learning outcomes

  • Understand the definition and properties of a tangent to a circle.
  • Differentiate between a tangent and a secant.
  • Identify the point of contact between a tangent and a circle.
  • Apply the Pythagorean theorem to find the length of a tangent segment.
  • Construct parallel tangents and secants to a given line with respect to a circle.

Topics covered

Paper topics

  • Tangents to a circle
  • Secants to a circle
  • Point of contact
  • Number of tangents
  • Parallel tangents
  • Radius perpendicular to tangent
  • Pythagorean theorem in tangent problems
  • Geometric constructions of tangents and secants

Important topics

  • Definition and properties of tangents
  • Radius perpendicular to tangent theorem
  • Calculating tangent length using Pythagoras theorem
  • Distinguishing between tangents and secants
  • Construction of parallel tangents

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Questions and Solutions

Question 1

How many tangents can a circle have?
Solution: A circle can have an infinite number of tangents. This is because a circle has infinitely many points on its circumference, and a unique tangent line can be drawn at each of these points. Each tangent line touches the circle at exactly one point.

Question 2

Fill in the blanks: (i) A tangent to a circle intersects it in _____ point(s). (ii) A line intersecting a circle in two points is called a ______. (iii) A circle can have _____ parallel tangents at the most. (iv) The common point of a tangent to a circle and the circle is called _____.
Solution: (i) A tangent to a circle intersects it in one point(s). This is the defining characteristic of a tangent. (ii) A line intersecting a circle in two points is called a Secant. A secant line passes through the interior of the circle. (iii) A circle can have two parallel tangents at the most. These parallel tangents are always drawn at the endpoints of a diameter and are equidistant from the center. (iv) The common point of a tangent to a circle and the circle is called point of contact. This is the single point where the tangent line touches the circle.

Question 3

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) \sqrt{119} cm.
Solution: We are given a circle with center O and radius OP = 5 cm. PQ is a tangent at point P, and OQ = 12 cm. A key property is that the radius to the point of contact is perpendicular to the tangent at that point. Therefore, triangle OPQ is a right-angled triangle with the right angle at P. We can use the Pythagorean theorem in \triangle OPQ: OQ^2 = OP^2 + PQ^2 Substitute the given values: 12^2 = 5^2 + PQ^2 144 = 25 + PQ^2 Now, solve for PQ^2: PQ^2 = 144 - 25 PQ^2 = 119 Taking the square root of both sides to find the length of PQ: PQ = \sqrt{119} Thus, the length of PQ is \sqrt{119} cm. The correct option is (D).

Question 4

Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Solution: 1. Start with a circle: Draw a circle with center O. Let's denote the radius as 'r'. 2. Draw a given line: Let AB be the given line. We need to draw two lines parallel to AB, one tangent and one secant. 3. Construct a perpendicular: Draw a line segment from the center O that is perpendicular to the line AB. Let this perpendicular intersect AB at point P. This line segment OP represents the shortest distance from the center to the line AB. 4. Locate points for parallel lines: * For the tangent: Extend the line segment OP in one direction. Let it intersect the circle at point X. The line drawn through X and parallel to AB will be the tangent. * For the secant: Extend the line segment OP in the same direction. Find a point Y on this extended line segment such that Y is inside the circle (i.e., the distance OY is less than the radius 'r'). The line drawn through Y and parallel to AB will be the secant. 5. Draw the parallel lines: * Draw a line CD passing through X such that CD is parallel to AB. CD is the tangent to the circle at point X. * Draw a line EF passing through Y such that EF is parallel to AB. EF is the secant to the circle, intersecting it at two points. Diagrammatic Representation: Imagine a circle. Draw a horizontal line AB below the circle. Draw a vertical line from the center O meeting AB at P. Extend this vertical line upwards to meet the circle at X. Draw a horizontal line CD through X; this is the tangent. Now, pick a point Y on the vertical line segment OP, between O and P. Draw another horizontal line EF through Y; this is the secant.

Common mistakes

  • Confusing a tangent with a secant.
  • Incorrectly applying the Pythagorean theorem.
  • Errors in geometric constructions.
  • Misunderstanding the infinite nature of tangents.

Revision tips

  • Review the definitions of tangent, secant, and point of contact.
  • Practice drawing diagrams accurately for each problem.
  • Memorize the theorem stating that the radius is perpendicular to the tangent at the point of contact.
  • Work through all solved examples and exercises to reinforce concepts.

Practice MCQs

Q1. How many tangents can a circle have at most?

Q2. A line intersecting a circle at two distinct points is called:

Q3. The common point of a tangent and the circle is known as the:

Q4. If a tangent PQ at point P on a circle of radius 5 cm has OQ = 12 cm (O is the center), what is the length of PQ?

Q5. How many parallel tangents can a circle have at the most?

Frequently asked questions

What is a tangent to a circle?

A tangent to a circle is a line that touches the circle at exactly one point, known as the point of contact. It does not pass through the interior of the circle.

What is the relationship between the radius and the tangent at the point of contact?

The radius drawn to the point of contact of a tangent is always perpendicular to the tangent itself.

How many tangents can be drawn to a circle?

A circle can have infinitely many tangents because there are infinitely many points on the circumference, and a unique tangent can be drawn at each point.

What is a secant?

A secant is a line that intersects a circle at two distinct points.

How can I find the length of a tangent segment?

If you know the distance from the center to the external point (hypotenuse) and the radius (one leg), you can use the Pythagorean theorem to find the length of the tangent segment (the other leg).

How do these solutions help in exam preparation?

These solutions provide clear, step-by-step explanations for all problems in Chapter 10, helping students understand concepts, practice problem-solving techniques, and build confidence for their exams.

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