Ninth Mathematics

Numbers which can be written in the form of `p/q` where p and q are integers and q≠0 are called RATIONAL NUMBERS.

All Natural Numbers, i.e. 1, 2, 3, 4, 5, . . . . can be written in the form of `p/q`, where, q≠0. Thus, all Natural Numbers are Rational Number.

All Whole Numbers, i.e. 0, 1, 2, 3, 4, . . . . can be written in the form of `p/q`, where, q≠0. Thus, every Whole Numbers is a Rational Number.

All Integers can be written in the form of `p/q` where, q≠0. Thus, all Integers are Rational Number.

When it is said that, `p/q` is a rational number, or when `p/q` is represented on the number line, it is assumed that q≠0 and p and q have no common factors other than 1. This, means p and q are co-prime. So on the Number Line, among the infinitely many fractions equivalent to `1/2`, but we will choose `1/2` to represent all of them . . .

Expressions having many terms are called POLYNOMIAL. The word "Polynomial" comes from two words "Poly" and "Nomen". "Poly" is a Greek word which means "many" and "Nomen" is a Latin word which means "Name or Term".

Thus, "Polynomial" = Poly + Nomial = Many + Term

An expression having variables and coefficients which involves the basic operations addition, subtraction and multiplication only and non-negative integer exponents of variables is called A POLYNOMIAL.

This means, an expression in which there are (a) variables and coefficients (b) consists of only three basic operation i.e. addition, multiplication and subtraction (c) and non-negative integer exponents of variables is considered as Polynomial. For example, 2, 2x, 3x, 3x + 2, x2 + x + 7, etc. are some examples of polynomials . . .

Line Segment A part of a line with two end points is called a Line Segment. A line segment is denoted by `bar(AB)`

Ray A part of a line with one end point is called a Ray. A ray is denoted by `vec(AB)`.

Collinear Points and Non-collinear Points If three or more points lie on the same line, they are called Collinear Points. Otherwise they are called Non-Collinear Points

Angle: When two rays originate from the same end point an angle is formed.

Arms and Vertex: The rays making an angle are called Arms while the point where both the arms meet is called the Vertex.

Types of Angle : On the basis of measurement angles can be divided into five types. These are Acute angle, Right angle, Obtuse angle, Straight angle and Reflex angle. . . .

NCERT Solutions

Solution of NCERT Exercise 7.1

Question (1) In quadrilateral ACBD, AC = AD and AB bisects ∠ A (see figure). Show that Δ ABC ≅ Δ ABD. What can you say about BC and BD?

Solution: Given, Quadrilateral ACBD, and AC = AD And AB bisects ∠ A

Then to prove: Δ ABC ≅ Δ ABD And BC and BD =?

Proof: In Δ ABC and Δ ABD. And AC = AD (Given). And ∠ CAB = ∠ BAD [Because according to question AB bisects ∠ A] . . .

A closed polygon with four sides, four angles and four vertices is called A QUADRILATERAL.

Lines joining opposite vertices of a quadrilateral are called DIAGONALS of quadrilateral. A quadrilateral can have maximum two diagonals.

Types of Quadrilateral: Quadrilaterals can be divided into following types:

(1) Parallelogram : A quadrilateral in which opposite sides parallel and equal is called A Parallelogram

Rectangle: A quadrilateral with opposite sides equal and all angles equal is called a rectangle. In other words a rectangle is a parallelogram in which every angle is a right angle.

(3) Square: A rectangle with all sides equal and all angles equal is called A Square. The length of both of the diagonals of a square is also equal. . .

What is a Cube?: Cube is a three dimensional object having six square faces. Each of the vertex of a cube has three meeting side. For example a dice. A dice is a typical example of a cube.

Since, a cube has six square faces, thus area of one square multiplied by 6 gives the area or surface area of a cube.

Now, Area of a square having edge equal to "a" = a²

Thus, Area of a cube = 6 × a²

Thus, Area or Total Surface Area or Lateral Surface Area of a Cube = 6 a²

Cuboid

A Cuboid is a three dimensional shaped object having six rectangular faces. For example a brick or a match box. Generally brick and a match box are typical examples of cuboid.

Cuboid means object having cube like shape. . . .