CBSE Class 10 Mathematics Chapter 10: Circles NCERT Solutions

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NCERT Solutions PDF Class 10 PDF

This chapter provides comprehensive NCERT Solutions for Class 10 Mathematics, focusing on Circles. Students will explore fundamental concepts related to circles, including tangents and secants. The solutions cover definitions, properties, and applications, such as determining the number of tangents a circle can have, understanding the relationship between tangents and points on a circle, and solving problems involving the lengths of tangents using the Pythagorean theorem. It also includes exercises on drawing parallel tangents and secants to a given line. These solutions are designed to help students grasp the geometric principles of circles and prepare effectively for their board examinations by offering clear, step-by-step explanations.

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Quick info

BoardCBSE
ClassClass 10
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 10

Chapter summary

Chapter 10, Circles, for Class 10 Mathematics NCERT Solutions, delves into the properties of circles and lines related to them. It covers the definitions and characteristics of tangents and secants, including the number of tangents possible from a point and the nature of lines intersecting a circle at one or two points. The exercises focus on understanding these concepts and applying them to solve problems, such as calculating the length of a tangent segment using the Pythagorean theorem and constructing parallel tangents and secants.

Learning outcomes

  • Understand the definition and properties of a tangent to a circle.
  • Identify the number of tangents that can be drawn to a circle.
  • Differentiate between a tangent and a secant.
  • 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
  • Number of tangents
  • Point of contact
  • Parallel tangents
  • Properties of tangents
  • Pythagorean theorem in tangent problems
  • Geometric constructions related to tangents

Important topics

  • Definition and properties of tangents
  • Relationship between radius and tangent at the point of contact
  • Calculating tangent length using Pythagoras theorem
  • Distinguishing between tangents and secants

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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 an infinite number of points on its circumference, and a unique tangent can be drawn at each of these points.

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). (ii) A line intersecting a circle in two points is called a Secant. (iii) A circle can have two parallel tangents at the most. (iv) The common point of a tangent to a circle and the circle is called point of contact.

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 Q is a point on the line through O such that OQ = 12 cm. According to a theorem, the radius drawn to the point of contact is perpendicular to the tangent at that point. Therefore, in triangle OPQ, the angle at P is a right angle (\angle OPQ = 90^{\circ}). We can use the Pythagorean theorem in the right-angled 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} cm. Thus, 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: Let's construct the required figure step-by-step: 1. Draw a circle: Start by drawing a circle with a center, let's call it O. Let the radius of the circle be 'r'. 2. Draw a given line: Draw any line, let's call it line AB. This is the line to which our other two lines will be parallel. 3. Draw a perpendicular from the center: Draw a line segment from the center O that is perpendicular to the line AB. Let this perpendicular intersect AB at point P. 4. Construct the tangent: To draw a tangent parallel to AB, we need a line that touches the circle at exactly one point and is parallel to AB. Draw a line through a point on the circle that is furthest from AB along the perpendicular line from O. Alternatively, draw a line parallel to AB passing through a point on the circle's circumference, such that it is perpendicular to the radius at that point. Let this line be CD. Line CD is a tangent to the circle and is parallel to AB. 5. Construct the secant: To draw a secant parallel to AB, we need a line that intersects the circle at two distinct points and is parallel to AB. Draw a line parallel to AB that passes through the interior of the circle, but not through the center in a way that it becomes a diameter line. This line will intersect the circle at two points. Let this line be EF. Line EF is a secant to the circle and is parallel to AB. In the resulting diagram, you will have a circle, the original line AB, a tangent line CD parallel to AB, and a secant line EF parallel to AB. The tangent CD touches the circle at one point, while the secant EF intersects the circle at two points.

Common mistakes

  • Confusing a tangent with a secant.
  • Incorrectly applying the Pythagorean theorem in tangent length calculations.
  • Misunderstanding the condition that the radius is perpendicular to the tangent at the point of contact.

Revision tips

  • Clearly define and differentiate between tangents and secants.
  • Memorize the theorem stating that the radius is perpendicular to the tangent at the point of contact.
  • Practice drawing diagrams accurately for geometry problems involving circles.
  • Work through the example problems to understand the application of theorems like Pythagoras.

Practice MCQs

Q1. How many tangents can be drawn to a circle from a point outside the circle?

Q2. A tangent to a circle intersects it in how many point(s)?

Q3. A line intersecting a circle in two distinct points is called a:

Q4. In a circle, what is the maximum number of parallel tangents that can be drawn?

Q5. The common point of a tangent and the circle is called the:

Frequently asked questions

What is a tangent to a circle?

A tangent to a circle is a line that intersects the circle at exactly one point. This point is called the point of contact.

How many tangents can a circle have?

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

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 perpendicular to the tangent. This property is crucial for solving problems using the Pythagorean theorem.

What is a secant to a circle?

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

How can we find the length of a tangent segment?

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

What is the maximum number of parallel tangents a circle can have?

A circle can have at most two parallel tangents. These tangents are drawn at the opposite ends of a diameter.

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