## Number System

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 Numbers**.

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

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

When it is said, 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 choose only 1/2 to represent all of them . . .

## Triangles-9th-Math

**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] . . .

## Polynomials

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 . . .

## Lines and Angles

**Line Segment:** A part of a line with two endpoints is called a **Line Segment**. A line segment is denoted by AB

**Ray** A part of a line with one endpoint is called a **Ray**. A ray is denoted by ^{→}/_{A}.

**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 endpoint 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. . . .

## Quadrilaterals: class 9 math

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. . .

## Surface Areas and Volumes

**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**" = a^{2}

Thus, **Area of a cube** = 6 × a^{2}

Thus, **Area or Total Surface Area or Lateral Surface Area of a Cube = 6 a ^{2}**

## 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. . . .