# NCERT Solutions for Class 9 Maths

NCERT Solutions for Class 9 Maths provides chapter-wise solutions to problems given in the respective NCERT textbook. The questions solved here in the solutions provided by GeeksforGeeks, give an approach to understand the concepts in a better way. This solution covers all the chapters in the NCERT book like Number System, Coordinate Geometry, Triangles, Circles, Mensuration, and many more with solutions for all the relevant exercises.

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### Chapter 1: Number Systems

The chapter Number systems briefly cover the concept of rational and irrational numbers, real numbers, and their representation on the number line. Along with the laws of rational exponents and integral powers. This chapter in total consists of six exercises, Exercise 1.1 helps to learn the basics of rational and irrational numbers, Exercise 1.2 explains the concept of real numbers and their expansion. Further, Exercise 1.3 helps to learn ‘How to represent real numbers on the number line’. Later, in exercises 1.4, 1.5, and 1.6 mathematical operations, rationalizing, and laws of rational exponents and integral powers for real numbers are discussed.

### Chapter 2: Polynomials

The chapter Polynomials guides to learn polynomials of degree 1 (Linear polynomials), 2 (quadratic polynomials), and 3 (cubic polynomials), etc, and terminology related to it. This chapter also covers the concept of factor theorem, algebraic identities of quadratic polynomials, split the middle theorem, algebraic identities for cubic polynomials, etc. In this chapter five exercises are there in which Exercises 2.1, and 2.2 cover topics like polynomials in one variable, zeroes of a polynomial, types of polynomials, and their degree. Exercises 2.3 and 2.4 are based on the concept of the factor theorem. Further, Exercise 2.5 helps to learn the algebraic identities of quadratic and cubic polynomials and split the middle theorem.

- Exercise 2.1
- Exercise 2.2
- Exercise 2.3
- Exercise 2.4
- Exercise 2.5 Set 1, Set 2

### Chapter 3: Coordinate Geometry

This chapter helps to learn the concepts of the Cartesian plane and various terminology related to it in detail. There are three exercises in this chapter, Exercise 3.1 explained How to locate a point on the coordinate plane?. While Exercises 3.2 and 3.3 discussed the topic ‘Plotting a point in the plane if its coordinates are given’.

### Chapter 4: Linear Equations in Two Variables

Linear equations in one variable have been already discussed in earlier classes which deals with unique solutions. Now, in this chapter linear equation comes with two variables like ax + by + c = 0. Exercises 4.1 and 4.3 are designed to learn about the formation and determination of the solution of a linear equation in two variables. However, Exercises 4.3 and 4.4 gave detail learning of the topic of the graph of a linear equation in two variables.

### Chapter 5: Introduction to Euclid’s Geometry

The chapter Introduction to Euclid’s Geometry explains Euclid’s method to geometry and linked it with modern day’s geometry. Euclid approached geometry by defining different axioms and postulates. This chapter contains two exercises, both Exercises 5.1 and 5.2 are based on the terms defined in the provided axioms and postulates.

### Chapter 6: Lines and Angles

As the name of the chapter says, this chapter deals with the properties of the angle formed when two or more lines intersect each other and the properties of these angles formed. This chapter consists of three exercises in which Exercises 6.1 and 6.2 give the knowledge about the different types of angles linear, reflex, right, obtuse, acute, complementary, and supplementary angles and their measures. Moreover, Exercise 6.3 covers the topic of angle sum property.

### Chapter 7: Triangles

This chapter gives a detailed description of the congruence of triangles, their rules, properties, and the inequalities in triangles. In total five exercises, the latter topics are covered. Exercises 7.1, 7.2, and 7.3 lay emphases on the topic of congruence of triangles and properties of triangles. Further, Exercises 7.4 and 7.5 are based on the triangle inequalities.

### Chapter 8: Quadrilateral

The chapter Quadrilateral is all about the different types of quadrilaterals (rectangle, square, parallelogram, trapezium, etc) and their different properties like the angle sum property of a quadrilateral. Further, a brief introduction of the midpoint theorem and its converse is provided in the last section of the chapter. This chapter consists of two exercises, in which exercise 8.1 helps to learn about the different properties of different quadrilaterals while exercise 8.2 explains the application of the mid-point theorem and its converse.

### Chapter 9: Areas of Parallelograms and Triangles

As the name suggests Areas of parallelogram and triangles, this chapter gives the understanding of the formula to determine the areas of different closed shapes using the relation between the areas of different geometric shapes. A total of four exercises present in the chapter in which Exercise 9.1 is based on the topic ‘Figures on the Same Base and Between the same parallels’. While Exercise 9.2 is based on the problems on the theorem for the parallelograms that have the same area and Exercises 9.3 and 9.4 cover the problems from the topic, triangles on the same base and between the same Parallels.

- Exercise 9.1
- Exercise 9.2
- Exercise 9.3 Set 1, Set 2
- Exercise 9.4

### Chapter 10: Circles

This chapter is related to the different properties and the terminology of the circles. Total 6 exercises are there in the chapter, Exercise 10.1 and 10.2 are related to the terminologies and the basic properties of the circles. However, Exercise 10.3 is based on the two theorems of the circles. Exercise 10.4 works upon the problem related to equal chords and their distances from the centre. Moreover, exercises 10.5 and 10.6 are based on the topics and theorems related to the angle subtended by an arc of a circle.

### Chapter 11: Constructions

The chapter Constructions in this class helps to understand the approach to construct different types of triangles for different given conditions. This chapter comprises of two exercises in which the problems in Exercise 11.1 deal with the construction of a different angle or the bisector of a given angle while, Exercise 11.2 are related to constructions of triangles, in the case when different parameters are given.

### Chapter 12: Heron’s Formula

This chapter discusses only Heron’s formula and its applications which are used to determine the area of a triangle when all the three sides of the triangles are provided. The chapter comprises of only two exercises, both Exercises 12.1 and 12.2 consist of problems related to the application of Heron’s formula.

### Chapter 13: Surface Areas and Volumes

The chapter Surface areas and volumes mainly deal with the formula, to determine the surface areas and volumes of different shapes like cube, cuboid, and cylinder and their applications. The problems related to each topic of this chapter are discussed in nine different exercises. Exercises 13.1, 13.2, and 13.3 concepts like the surface areas and volume of the various geometrical shapes like cube, cuboid, and more. However, Exercises 13.4, 13.5, 13.6, 13.7, 13.8, and 13.9 are focused on the problems based on the applications of the mentioned formulae.

- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
- Exercise 13.4
- Exercise 13.5
- Exercise 13.6
- Exercise 13.7
- Exercise 13.8
- Exercise 13.9

### Chapter 14: Statistics

This chapter helps to learn the representation of data in a different manner along with the frequency distribution. With the help of total four exercises the concept of graphical representation of data, using various graphs like Frequency polygons, Histograms, Bar graphs, etc. Exercise 14.1 comprises of the descriptive problems to understand the basics of statistics and Exercise 14.2 helps to learn to plot the frequency distribution table for the given data. While, Exercise 14.3 include problems on frequency polygons and histogram, and Exercise 14.4 is based on the concept of mean, mode, and median.

### Chapter 15: Probability

The last chapter Probability discusses the concept of the Chances of occurrence of a specific outcome in an experiment. This chapter consists of only one exercise which is based on the problems from everyday life examples to understand the experimental approach to probability.