CBSE Class 10 Maths Chapter 10: Circles NCERT Solutions

Reviewed by · Updated on

NCERT Solutions PDF Class 10 PDF

This chapter provides essential NCERT Solutions for Class 10 Mathematics, focusing on Circles. It covers fundamental concepts related to circles, including tangents and secants. Students will learn about the properties of tangents, such as the number of tangents a circle can have and the relationship between the radius and the tangent at the point of contact. The solutions also explain the definition of a secant and how to draw parallel tangents and secants to a given line. These solutions are designed to help students understand the geometric properties of circles and prepare effectively for their board examinations by offering clear, step-by-step explanations and accurate answers.

Home

Quick info

BoardCBSE
ClassClass 10
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 10: Circles

Chapter summary

Chapter 10, Circles, for Class 10 Maths NCERT Solutions delves into the properties of tangents and secants. It addresses key concepts like the number of tangents possible, the definition of a tangent and secant, and the relationship between parallel tangents. The chapter includes problems requiring the application of the Pythagorean theorem to find the length of a tangent segment and exercises involving the construction of parallel tangents and secants. These solutions provide a clear understanding of circle geometry.

Learning outcomes

  • Understand the definition and properties of a tangent to a circle.
  • Determine the number of tangents a circle can have.
  • Identify a secant and differentiate it from a tangent.
  • Apply the Pythagorean theorem to solve problems involving tangents.
  • Construct parallel tangents and secants to a given line.
  • Understand the concept of the point of contact.

Topics covered

Paper topics

  • Tangents to a circle
  • Secants to a circle
  • Number of tangents
  • Point of contact
  • Parallel tangents
  • Parallel secants
  • Pythagorean theorem in circle geometry
  • Geometric constructions related to circles

Important topics

  • Definition and properties of tangents
  • Relationship between radius and tangent
  • Calculating tangent length using Pythagorean theorem
  • Distinguishing between tangents and secants
  • Construction of parallel tangents and secants

PDF preview

Read page by page below. PDF is streamed from the official NCERT website — no download button on this page.

Loading document …
Page of
Loading page …

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. Each point on the circle represents a potential point of contact for a tangent line.

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 single point is known as the point of contact. (ii) A line intersecting a circle in two points is called a Secant. (iii) A circle can have two parallel tangents at the most. These tangents would be drawn at opposite ends of a diameter. (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 OQ = 12 cm. According to a fundamental theorem in circle geometry, the radius drawn 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, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In \triangle OPQ:

OQ^2 = PQ^2 + OP^2

Substitute the given values:

12^2 = PQ^2 + 5^2 144 = PQ^2 + 25

Now, solve for PQ^2:

PQ^2 = 144 - 25 PQ^2 = 119

To find the length of PQ, take the square root of both sides:

PQ = \sqrt{119}

Thus, the length of the tangent segment PQ is \sqrt{119} cm. This corresponds to option (D).

Answer: (D) \sqrt{119} cm.

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. Mark the radius, say 'r'. 2. Draw a given line: Draw any straight line, let's call it AB. This is the line to which our tangent and secant 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: Draw a line that is parallel to AB and passes through the point where the perpendicular from O intersects the circle on one side. Let this point be X. The line through X, parallel to AB, will be tangent to the circle at X. Let's call this tangent line CD. 5. Construct the secant: Draw another line that is parallel to AB and intersects the circle at two distinct points. This line should be positioned between the center O and the tangent CD. Let this line intersect the circle at points Y and Z. The line passing through Y and Z, parallel to AB, will be the required secant. Let's call this secant line EF. In this construction, line CD is tangent to the circle (intersects at one point X), and line EF is a secant to the circle (intersects at two points Y and Z). Both CD and EF are parallel to the given line AB.
Diagram showing a circle with a tangent and a secant parallel to a given line.

Common mistakes

  • Confusing a tangent with a secant.
  • Incorrectly applying the Pythagorean theorem.
  • Errors in geometric constructions.
  • Misunderstanding the relationship between radius and tangent.

Revision tips

  • Review the definitions of tangent, secant, and point of contact.
  • Practice drawing diagrams accurately for each problem.
  • Ensure you understand the theorem stating that the radius is perpendicular to the tangent at the point of contact.
  • Work through the example problems to solidify your understanding of applying theorems.

Practice MCQs

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

Q2. A tangent to a circle intersects it in how many points?

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

Q4. In the given figure, if OP = 5 cm and OQ = 12 cm, what is the length of PQ?

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

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

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 of a circle, and a tangent can be drawn at each point.

What is the difference between a tangent and a secant?

A tangent intersects a circle at exactly one point, while a secant intersects a circle at two distinct points.

How is the Pythagorean theorem used with tangents?

When a tangent segment is drawn from an external point to the point of contact, and a line is drawn from the center to the external point, a right-angled triangle is formed. The Pythagorean theorem can be applied to find unknown lengths.

What does it mean for a tangent to be parallel to a given line?

It means that the tangent line and the given line never intersect, no matter how far they are extended, and maintain a constant distance between them. A circle can have at most two parallel tangents.

How can these NCERT solutions help in exam preparation?

These solutions provide clear, step-by-step explanations for each question, helping students understand the concepts and methods required to solve problems related to circles, tangents, and secants, thereby aiding in effective exam revision.

Important links

Content reviewed by the NCERT Help team. Editorial Team and update policy

NCERT Solutions PDF PDF on NCERT Help. URL unchanged for search indexing.