# NCERT Solutions for Class 10 Maths Chapter 10 Circles

NCERT Solutions for Class 10 Maths Chapter 10 Circles-

In our previous classes, we have learnt that a circle is a closed shape with a collection of points in a plane which are at a specific distance ( called radius) from a fixed point (called centre). We have also studied important terms related to the circle like segment, arc, sector, chord etc. In this chapter, there are two exercises with 17 questions in them. The NCERT Solutions for Class 10 Maths Chapter 10 Circles are solved by our maths experts to help students in their preparation of CBSE Class 10 Board exam. These NCERT solutions are the step-by-step explanation of each and every question of class 10 NCERT textbook.

In this chapter, we will study the different conditions that arise when a line and a circle are given in a plane. And to solve these type of situations we will learn the concept of the tangent to a circle and the number of tangents from a point on a circle in NCERT Class 10 Maths Chapter 10 Circles. This chapter has fundamental concepts which are important for students in their future studies. This chapter introduces some complex and important terms like tangents, tangents to a circle, number of tangents from a point on a circle. NCERT Class 10 Maths Chapter 10 Circles is a very interesting chapter due to the involvement of geometrical calculations and the diagrams.

The two important theorems of Chapter 10 Circles are-

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact.

Proof: Assume O is the centre of the circle and XY is the tangent to the circle at point P. And we need to prove that OP is perpendicular to the XY.

Take point Q on XY other than point P and join OQ. The point Q lies outside the circle. Therefore, OQ is longer than the radius of the circle OP.

$OQ>OP$

Except for point P, this happens for every time on the line XY. OP is the shortest distances of the point O to the points of line XY. So that's why OP is perpendicular to the line XY.

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

Proof: Assume O is the centre of the circle, a point P lying outside a circle and two tangents PQ and PR on the circle from point P. We need to prove $\dpi{100} PQ = PR$.

For this, we join OP, OQ and OR. Then $\dpi{100} \angle OQP$ and $\dpi{100} \angle ORP$ are right angles, because these two are angles between the tangents and radii, and according to Theorem 10.1, they are right angles. Now in right triangles ORP and OQP.

$\dpi{100} \\OQ =OR\:\:\:\:\:\:\:\:\:\:\:\:\;\:\:\:\:\:\:\:\:\:\:\:\:\;(Radii \:of\: the\: same\: circle)\\\\OP =OP\:\:\:\:\:\:\:\:\:\:\:\:\;\:\:\:\:\:\:\:\:\:\:\:\:\;(Common)\\\\\bigtriangleup OQP \cong \bigtriangleup ORP\:\:\:\:\:\:\:\:\:\:\: (RHS)\\\\PQ =PR\:\:\:\:\:\:\:\:\:\:\: \:\:\:\:\:\:\:\:\:\:\: \:\:\:\:\:(CPCT)$

## 10.1 Introduction

10.2 Tangent to a Circle

10.3 Number of Tangents from a Point on a Circle

10.4 Summary

The complete Solutions of NCERT Class 10 Mathematics Chapter 10 is provided below:

## NCERT Solutions for Class 10 Maths - Chapter-wise

 Chapter No. Chapter Name Chapter 1 Real Numbers Chapter 2 Polynomials Chapter 3 Pair of Linear Equations in Two Variables Chapter 4 Quadratic Equations Chapter 5 Arithmetic Progressions Chapter 6 Triangles Chapter 7 Coordinate Geometry Chapter 8 Introduction to Trigonometry Chapter 9 Some Applications of Trigonometry Chapter 11 Constructions Chapter 12 Areas Related to Circles Chapter 13 Surface Areas and Volumes Chapter 14 Statistics Chapter 15 Probability

## NCERT Solutions for Class 10 - Subject Wise

 NCERT Solutions for Class 10 Maths NCERT Solutions for Class 10 Science