# Class 9th Math Ncert Solution

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## Chapter-wise NCERT Solutions for Class 9 Maths

### NCERT Solutions for Class 9 Maths Chapter 1 Number System

### NCERT Solutions for Class 9 Maths Chapter 2 Polynomial

(x + y)2 = x2 + 2xy + y2

(x – y)2 = x2 – 2xy + y2

x2 – y2 = (x + y) (x – y)

and their use in factorisation. In this chapter, we shall start our study with a particular type of algebraic expression, called polynomial, and the terminology related to it. We shall also study the Remainder Theorem and Factor Theorem and their use in the factorisation of polynomials. In addition to the above, we shall study some more algebraic identities and their use in factorisation and in evaluating some given expressions

### NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry

The chapter Coordinate Geometry includes the concepts of the Cartesian plane, coordinates of a point in xy – plane, terms, notations associated with the coordinate plane, including the x-axis, y-axis, x- coordinate, y-coordinate, origin, quadrants and more. Students, in this chapter, will also be studying the concepts of Abscissa and ordinates of a point as well as plotting and naming a point in xy – plane. There are 3 exercises in this chapter that contain questions revolving around the topics mentioned in the chapter, helping the students get thorough with the concepts.

Class 9 Maths NCERT Solutions Chapter 3 Exercises |
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Exercise 3.1 – 2 Questions (1 short answer, 1 long answer) |

Exercise 3.2 – 2 Questions (2 short answers) |

Exercise 3.3 – 2 Questions (1 short answer, 1 long answer) |

### NCERT Solutions for Class 9 Maths Chapter 4 Linear Equation in one variable

In earlier classes, you have studied linear equations in one variable. Can you write down a linear equation in one variable? You may say that x + 1 = 0, x + 2 = 0 and 2 y + 3 = 0 are examples of linear equations in one variable. You also know that such equations have a unique (i.e., one and only one) solution. You may also remember how to represent the solution on a number line. In this chapter, the knowledge of linear equations in one variable shall be recalled and extended to that of two variables. You will be considering questions like: Does a linear equation in two variables have a solution? If yes, is it unique? What does the solution look like on the Cartesian plane? You shall also use the concepts you studied in Chapter 3 to answer these questions.

Class 9 Maths NCERT Solutions Chapter 4 Exercises |
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Exercise 4.1 – 2 Questions (2 short answers) |

Exercise 4.2 – 4 Questions (3 short answers, 1 long answer ) |

Exercise 4.3 – 8 Questions (4 short answers, 4 long answers) |

Exercise 4.4 – 2 Questions (2 long answers) |

### NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclids Geometry

(Axiom) 1. Given two distinct points, there exists one and only one line through them.

(Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common.

**Important Axioms and Postulates – **

Some of Euclid’s axioms are:

(1) Things which are equal to the same thing are equal to one another.

(2) If equals are added to equals, the wholes are equal.

(3) If equals are subtracted from equals, the remainders are equal.

(4) Things which coincide with one another are equal to one another.

(5) The whole is greater than the part.

(6) Things which are double of the same things are equal to one another.

(7) Things which are halves of the same things are equal to one another.

**Euclid’s Five Postulates**

Postulate 1 – A straight line may be drawn from any one point to any other point.

Postulate 2 – A terminated line can be produced indefinitely.

Postulate 3 – A circle can be drawn with any centre and any radius.

Postulate 4 – All right angles are equal to one another.

Postulate 5 – If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Class 9 Maths NCERT Solutions Chapter 5 Exercises |
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Exercise 5.1 – 7 Questions (4 short answers, 3 long answers) |

Exercise 5.2 – 2 Questions (1 short answer, 1 long answer) |

### NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

In Chapter 5, you have studied that a minimum of two points are required to draw a line. You have also studied some axioms and, with the help of these axioms, you proved some other statements. In this chapter, you will study the properties of the angles formed when two lines intersect each other, and also the properties of the angles formed when a line intersects two or more parallel lines at distinct points. Further you will use these properties to prove some statements using deductive reasoning (see Appendix 1). You have already verified these statements through some activities in the earlier classes. In your daily life, you see different types of angles formed between the edges of plane surfaces. For making a similar kind of model using the plane surfaces, you need to have a thorough knowledge of angles. For instance, suppose you want to make a model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you would make it? You would keep some of the sticks parallel to each other, and some sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried building, she has to draw intersecting lines and parallel lines at different angles. Without the knowledge of the properties of these lines and angles, do you think she can draw the layout of the building? In science, you study the properties of light by drawing the ray diagrams. For example, to study the refraction property of light when it enters from one medium to the other medium, you use the properties of intersecting lines and parallel lines. When two or more forces act on a body, you draw the diagram in which forces are represented by directed line segments to study the net effect of the forces on the body. At that time, you need to know the relation between the angles when the rays (or line segments) are parallel to or intersect each other. To find the height of a tower or to find the distance of a ship from the light house, one needs to know the angle formed between the horizontal and the line of sight. Plenty of other examples can be given where lines and angles are used. In the subsequent chapters of geometry, you will be using these properties of lines and angles to deduce more and more useful properties

**Important Points –**

- If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and viceversa. This property is called as the Linear pair axiom.
- If two lines intersect each other, then the vertically opposite angles are equal.
- If a transversal intersects two parallel lines, then

(i) each pair of corresponding angles is equal,

(ii) each pair of alternate interior angles is equal,

(iii) each pair of interior angles on the same side of the transversal is supplementary.

- If a transversal intersects two lines such that, either

(i) any one pair of corresponding angles is equal, or

(ii) any one pair of alternate interior angles is equal, or

(iii) any one pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.

Class 9 Maths NCERT Solutions Chapter 6 Exercises |
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Exercise 6.1 – 6 Questions (5 short answers, 1 long answer) |

Exercise 6.2 – 6 Questions (3 short answers, 3 long answers) |

Exercise 6.3 – 6 Questions (5 short answers, 1 long answer) |

### NCERT Solutions for Class 9 Maths Chapter 7 Triangles

You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a triangle. (‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ∆ ABC (see Fig. 7.1); AB, BC, CA are the three sides, ∠ A, ∠ B, ∠ C are the three angles and A, B, C are three vertices. In Chapter 6, you have also studied some properties of triangles. In this chapter, you will study in details about the congruence of triangles, rules of congruence, some more properties of triangles and inequalities in a triangle. You have already verified most of these properties in earlier classes. We will now prove some of them

**Important Axioms and Theorems –**

**Axiom 7.1 (SAS congruence rule) –** Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.

**Theorem 7.1 (ASA congruence rule) –** Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.

**Theorem 7.2 –** Angles opposite to equal sides of an isosceles triangle are equal.

**Theorem 7.3 –** The sides opposite to equal angles of a triangle are equal.

**Theorem 7.4 (SSS congruence rule) –** If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

**Theorem 7.5 (RHS congruence rule) –** If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.

**Theorem 7.6 –** If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater).

**Theorem 7.7 –** In any triangle, the side opposite to the larger (greater) angle is longer

**Theorem 7.8 –** The sum of any two sides of a triangle is greater than the third side.

### NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

**Important Theorems –**

A quadrilateral has four sides, four angles and four vertices.

**Angle Sum Property of a Quadrilateral – **The sum of the angles of a quadrilateral is 360o.

**Theorem 8.1 –** A diagonal of a parallelogram divides it into two congruent triangles

**Theorem 8.2 –** In a parallelogram, opposite sides are equal.

**Theorem 8.3 –** If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

**Theorem 8.4 –** In a parallelogram, opposite angles are equal.

**Theorem 8.5 –** If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

**Theorem 8.6 –** The diagonals of a parallelogram bisect each other.

**Theorem 8.7 –** If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

**Theorem 8.8 –** A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

**Theorem 8.9 –** The line segment joining the mid-points of two sides of a triangle is parallel to the third side.

**Theorem 8.10 –** The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.

Class 9 Maths NCERT Solutions Chapter 8 Exercises |
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Exercise 8.1 – 12 Questions (2 short answers, 6 long answers, 4 very long answers) |

Exercise 8.2 – 7 Questions (2 short answers, 2 long answers, 3 very long answers) |

### NCERT Solutions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles

**Topics Covered in Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles:**

**Important Theorems –**

Class 9 Maths NCERT Solutions Chapter 9 Exercises |
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Exercise 9.1 – 1 Question (1 short answer) |

Exercise 9.2 – 6 Questions (5 short answers, 1 long answer) |

Exercise 9.3 – 16 Questions (12 short answers, 4 long answer) |

Exercise 9.4 – 8 Questions (4 short answer, 1 long answer, 3 very long answers) |

### NCERT Solutions for Class 9 Maths Chapter 10 Circles

**Topics Covered in Class 9 Maths Chapter 10 Circles:**

Class 9 Maths NCERT Solutions Chapter 10 Exercises |
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Exercise 10.1 – 2 Questions (2 short answers) |

Exercise 10.2 – 2 Questions (2 long answers) |

Exercise 10.3 – 3 Questions (3 long answers) |

Exercise 10.4 – 6 Questions (6 long answers) |

Exercise 10.5 – 12 Questions (12 long answers) |

Exercise 10.6 – 10 Questions (10 long answers) |

### NCERT Solutions for Class 9 Maths Chapter 11 Constructions

**Topics Covered in Class 9 Maths Chapter 11 Constructions :**

**Important Points –**

Class 9 Maths NCERT Solutions Chapter 11 Exercises |
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Exercise 11.1 – 5 Questions (2 short answers, 2 long answers, 1 very long answer) |

Exercise 11.2 – 5 Questions (5 very long answer) |

### NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula

Class 9 Maths NCERT Solutions Chapter 12 Exercises |
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Exercise 12.1 – 6 Questions (2 short answers, 2 long answers, 2 very long answers) |

Exercise 12.2 – 9 Questions (4 long answers, 5 very long answers) |

### NCERT Solutions for Class 9 Maths Chapter 13 Surface Areas and Volumes

In this chapter, students shall learn to find the surface areas and volumes of cuboids and cylinders in detail and broaden the study to some other solids, such as cones and spheres. This chapter is just an extended version of the chapter mensuration, in which the students learnt about the surface areas and volumes in earlier classes. There are 8 exercises in this chapter, and these exercises contain problems that are based on surface areas and volumes of different solids such as cubes, cuboids, spheres, cylinders, cones, and hemispheres.

*Topics Covered in Class 9 Maths Chapter 13 Surface Areas and Volumes :*

*Topics Covered in Class 9 Maths Chapter 13 Surface Areas and Volumes :*

Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right circular

cylinders/cones.

**Important Formulas –**

Surface Area of a Cuboid = 2(lb + bh + hl)

where l, b and h are respectively the three edges of the cuboid

Surface Area of a Cube = 6a2

where a is the edge of the cube.

Curved Surface Area of a Cylinder = 2πrh

where r is the radius of the base of the cylinder and h is the height of the cylinder

Total Surface Area of a Cylinder = 2πr(r + h)

where h is the height of the cylinder and r its radius

Curved Surface Area of a Cone = 1/2 × l × 2πr = πrl

where r is its base radius and l its slant height

Total Surface Area of a Cone = πrl + πr2 = πr(l + r)

Surface Area of a Sphere = 4 π r2

where r is the radius of the sphere.

Curved Surface Area of a Hemisphere = 2πr2

where r is the radius of the sphere of which the hemisphere is a part.

Total Surface Area of a Hemisphere = 3πr2

Volume of a Cuboid = base area × height = length × breadth × height

Volume of a Cube = edge × edge × edge = a3

Volume of a Cylinder = πr2h

where r is the base radius and h is the height of the cylinder.

Volume of a Cone = 1/3 πr2h

where r is the base radius and h is the height of the cone.

Volume of a Sphere = 4/3 πr3

where r is the radius of the sphere.

Volume of a Hemisphere = 2/3 πr3

where r is the radius of the hemisphere.

Class 9 Maths NCERT Solutions Chapter 13 Exercises |
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Exercise 13.1 – 8 questions |

Exercise 13.2 – 11 questions |

Exercise 13.3 – 8 questions |

Exercise 13.4 – 9 questions |

Exercise 13.5 – 9 questions |

Exercise 13.6 – 8 questions |

Exercise 13.7 – 9 questions |

Exercise 13.8 – 10 questions |

Exercise 13.9 – 3 questions |

### NCERT Solutions for Class 9 Maths Chapter 14 Statistics

**Topics Covered in Class 9 Maths Chapter 14 Statistics :**

**Important Formulas –**

Class 9 Maths NCERT Solutions Chapter 14 Exercises |
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Exercise 14.1 – 2 questions (2 short answers) |

Exercise 14.2 – 2 questions (2 short answers) |

Exercise 14.3 – 9 questions (6 short answers, 3 long answers) |

Exercise 14.4 – 6 questions (2 short answers, 4 long answers) |

### NCERT Solutions for Class 9 Maths Chapter 15 Probability

**Topics Covered in Class 9 Maths Chapter 15 Probability :**

**Important Formulas –**

Class 9 Maths NCERT Solutions Chapter 15 Exercises |
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Exercise 15.1 – 10 Question ( 4 short answers, 3 long answers, 3 very long answers) |