Synopsis – Grade 9 Math Term I

Chapter 1: Number Systems

Natural numbers

The counting numbers 1, 2, 3 … are called natural numbers. The set of natural numbers is

denoted by N.

N = {1, 2, 3…}

Whole numbers

If we include zero to the set of natural numbers, then we get the set of whole numbers.

The set of whole numbers is denoted by W.

W = {0, 1, 2…}

Integers

The collection of numbers … –3, –2, –1, 0, 1, 2, 3 … is called integers. This collection is

denoted by Z, or I.

Z = {…, –3, –2, –1, 0, 1, 2, 3…}

Rational numbers

Rational numbers are those which can be expressed in the formp

q, where p, q are integers

and q 0.

Example:1 3 6

, , ,2 4 9

etc.

Every rational number „x ‟can be expressed asa

xb

, where a, b are integers such that

the HCF of a and b = 1 and b 0.

Every natural number, whole number or integer is a rational number.

There are infinitely many rational numbers between any two given rational numbers.

Example:

Find a rational number between 3

8 and

5

12.

Solution:

The mean of two given rational numbers gives a rational number between them.

Now, 3 5 19

8 12 24

A rational number between 3

8 and

5

12 =

3 5

198 12

2 48

Irrational numbers

Irrational numbers are those which cannot be expressed in the formp

q, where p, q are

integers and q 0.

Example: π, 2, 7, 14,0.0202202220.......

There are infinitely many irrational numbers.

π = 3.141592… is irrational. Its approximate value is assumed as 22

7 or as 3.14, both

of which are rational.

Real numbers

The collection of all rational numbers and irrational numbers is called real numbers.

A real number is either rational or irrational.

Every real number is represented by a unique point on the number line (and vice

versa).

So, the number line is also called the real number line.

Example:

Locate 6 on the number line.

Solution:

(a) Mark O at 0 and A at 2 on the number line, and then draw AB of unit length

perpendicular to OA. Then, by Pythagoras Theorem, OB 5

(b) Construct BD of unit length perpendicular to OB. Thus, by Pythagoras

theorem, 2

OD 5 12 6

(c) With centre O and radius OD, draw an arc intersecting the number line at point

P. Thus, P corresponds to the number 6 .

Real numbers and their decimal expansions

The decimal expansion of a rational number is either terminating or non-terminating

recurring (repeating).

Example:

151.875 Terminating

8

41.333....... 1.3 Non – terminating recurring  

3

A number whose decimal expansion is terminating or non-terminating repeating is

rational.

The decimal expansion of an irrational number is non-terminating non-recurring.

Moreover, a number whose decimal expansion is non-terminating non- recurring is

irrational.

Example:

2.645751311064……. is an irrational number

Representation of real numbers on the number line

Example: 3.32 can be visualize by the method of successive magnification on the number

line as follows:

Operation on real numbers

The sum or difference of a rational number and an irrational number is always

irrational.

The product or quotient of a non-zero rational number with an irrational number is

always irrational.

If we add, subtract, multiply or divide two irrational numbers, then the result may be

rational or irrational.

Identities

If a and b are positive real numbers, then

ab a b

a a

b b

2a b a b a b

2a b a b a b

a b c d ac ad bc bd

2

2a b a ab b

Rationalisation of denominator

The denominator of a b

x y

can be rationalised by multiplying both the numerator and

the denominator by x y , where a, b, x, y are integers.

Laws of exponents

Let a > 0 is a real number and p, q are rational numbers.

.p q p qa a a

q

p pqa a

p

p q

q

aa

a

p p pab a b

1

x xa a , where n is a positive integer.

Chapter 2: Polynomials

Polynomial in one variable

A polynomial p(x) in one variable i.e., x is an algebraic expression in x of the form

1

1 1 0..... ,n n

n np x a x a x a x a

where 0 1, ... na a a are constants and 0na .

0 1, ... na a a are the respective coefficients of 0 1 2, , ... nx x x x and n is called the degree of

the polynomial. 1

1 0, ...n n

n na x a x a

and 0 0a are called the terms of p(x).

Constant polynomial: A constant polynomial is of the form p x k , where k is a real

number. For example, –9, 10, 0 are constant polynomials.

The degree of a non-zero constant polynomial is zero.

Zero polynomial: A constant polynomial „0‟ is called zero polynomial.

The degree of a zero polynomial is not defined.

Classification of polynomials according to terms

A polynomial comprising one term is called a monomial, e.g., 3x, 5, 25t3.

A polynomial comprising two terms is called a binomial, e.g., 2t – 6, 3x4 + 2x etc.

A polynomial comprising three terms is called a trinomial, e.g., 3 65 2, 9.x x y y

Classification of polynomial according to their degrees

A polynomial of degree one is called a linear polynomial, e.g., 3x+ 2, 4x, x + 9.

A polynomial of degree two is called a quadratic polynomial, e.g., 2 9x , 23 4 6x x .

A polynomial of degree three is called a cubic polynomial, e.g., 3 310 3, 9x x .

Zeroes of a polynomial: A real number is said to be the zero of polynomial p x if

0p . In this case, is also called the root of the equation 0p x

A non-zero constant polynomial has no zeroes

Every real number is a zero of the zero polynomial

The maximum number of zeroes of a polynomial is equal to the degree of the

polynomial

A polynomial can have more than one zeroes

Example: Find the value of polynomial 33 2 9p x x x at 2x .

Solution:

3

3

3 2 9

2 3 2 2 2 9

24 4 9 19

p x x x

p

Thus, x = –2 is not the zero of the polynomial.

Division of a polynomial by another polynomial

If p(x) and g(x) are two polynomials such that degree of p(x) degree of g x and

0g x , then we can find polynomials q x and r x such that

p x g x q x r x , where 0r x or degree of r x < degree of g(x)

Here, p(x) is the dividend, g x is the divisor, q x is the quotient, and r x is the

remainder.

Example: Divide 4 3 22 2 7 15x x x x by x – 2.

Solution:

3

4 3 2

4 3

2

2

2 3

2 2 2 7 15

2

2 7 15

2 4

3 15

3 6

9

x x

x x x x x

x x

x x

x x

x

x

It can be easily verified that 4 3 2 32 2 7 15 2 2 3 9x x x x x x x .

Remainder theorem

If p(x) is a polynomial of degree greater than or equal to one and a is a real number, then,

when p(x) is divided by linear polynomial x – a, the remainder is p(a).

Factor theorem: If p(x) is a polynomial of degree x 1and a is any real number, then

x – a is a factor of p(x), if p(a) = 0

0p a , if x a is a factor of p(x)

Factorisation of polynomials: Polynomials can be factorised by using the algebraic

identities given below.

2 2 22x y x xy y

2 2 22x y x xy y

2 2x y x y x y

2x a x b x a b x ab

2 2 2 2 2 2 2x y z x y z xy yz zx

3 3 3 3 3 2 23 3 3x y x y xy x y x y x y xy

3 3 3 3x y x y xy x y 3 3 2 23 3x y x y xy

3 3 3 2 2 23x y z xyz x y z x y z xy yz zx

For example: Factorise 2 24 20 25x xy y

2 24 20 25x xy y 2 2

2 2 2 5 5x x y y

2 22 22 5 2

2 5 2 5

x y a ab b a b

x y x y

Chapter 3: Coordinate Geometry

To identify the position of an object or a point in a plane, we require two perpendicular

lines: one of them is horizontal and the other is vertical.

Cartesian system

A Cartesian system consists of two perpendicular lines: one of them is horizontal and

the other is vertical.

The horizontal line is called the x- axis and the vertical line is called the y -axis.

XOX is called the x-axis; YOY is called the y-axis

The point of intersection of the two lines is called origin, and is denoted by O.

OX and OY are respectively called positive x-axis and positive y-axis.

Positive numbers lie on the directions of OX and OY.

OX and OY are respectively called negative x-axis and negative y-axis.

The axes divide the plane into four equal parts.

The four parts are called quadrants, numbered I, II, III and IV, in anticlockwise from

positive x-axis, OX.

The plane is also called co-ordinate plane or Cartesian plane or xy -plane.

The coordinates of a point on the coordinate plane can be determined by the

following conventions.

The x-coordinate of a point is its perpendicular distance from the y-axis, measured

along the x-axis (positive along the positive x-axis and negative along the negative x-

axis).

The x-coordinate is also called the abscissa.

The y-coordinate of a point is its perpendicular distance from the x-axis, measured

along the y-axis ( positive along the positive y-axis and negative along the negative y -

axis)

The y-coordinate is also called the ordinate.

In stating the coordinates of a point in the coordinate plane, the x-coordinate comes

first and then the y-coordinate. The coordinates are placed in brackets.

If x = y, then (x, y) = ( y, x); and (x, y) (y, x) if x y.

The coordinates of the origin are (0, 0). Since the origin has zero distance from both

the axes, its abscissa and ordinate are both zero.

The coordinates of the point on the x-axis are of the form (a, 0) and the coordinates of

the point on the y-axis are of the form (0, b), where a, b are real numbers.

Example: What are the coordinates of points A and C in the given figure?

Solution:

It is observed that

x-coordinate of point A is 5

y-coordinate of point A is 2

Coordinates of point A are (5, 2)

x-coordinate of point C is –5

y-coordinate of point C is 2

Coordinates of point C are (–5, 2)

Relationship between the signs of the coordinates of a point and the quadrant of the

point in which it lies:

The 1st quadrant is enclosed by the positive x-axis and positive y-axis. So, a point in

the 1st quadrant is in the form (+, +).

The 2nd

quadrant is enclosed by the negative x-axis and positive y-axis. So, a point in

the 2nd

quadrant is in the form (–, +).

The 3rd

quadrant is enclosed by the negative x-axis and the negative y-axis. So, the

point in the 3rd

quadrant is in the form (–, –).

The 4th

quadrant is enclosed by the positive x-axis and the negative y-axis. So, the

point in the 4th

quadrant is in the form (+, –).

Location of a point in the plane when its coordinates are given

Example: Plot the following ordered pairs of numbers (x, y) as points in the coordinate

plane.

x –3 4 –3 0

y 4 –3 –3 2

Solution:

These points can be located in the coordinate plane as:

Chapter 5: Introduction to Euclid’s Geometry

Introduction to Euclid’s geometry

During Euclid‟s period, the notions of points, line, plane (or surface), and so on were

derived from what was seen around them.

Euclid’s definitions

Some definitions given in his book I of the „Elements‟ are as follows.

A point is that which has no part.

A line is breadth-less length.

A straight line is a line which lies evenly with the points on itself.

A surface is that which has length and breadth only.

The edges of a surface are lines.

A plane surface is a surface which lies evenly with the straight lines on itself.

In the above definitions, we can observe that some of the terms such as part, breadth,

length, etc. require better explanations.

Therefore, to define one thing, we require defining many other things and we may obtain

a long chain of definitions without an end. For such reasons, mathematicians agreed to

leave some geometric terms such as point, line, and plane undefined.

Euclid’s axioms and postulates

Axioms and postulates are the assumptions that are obvious universal truths, but are not

proved. Euclid used the term “postulate” for the assumptions that were specific to

geometry whereas axioms are used throughout mathematics and are not specifically

linked to geometry.

Some of Euclid’s axioms

Things that are equal to the same things are equal to one another.

If equals are added to equals, then the wholes are also equal.

If equals are subtracted from equals, then the remainders are equal.

Things that coincide with one another are equal to one another.

The whole is greater than the part.

Things that are double of the same things are equal to one another.

Things that 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.

Euclid has frequently assumed this postulate, without mentioning that there is a unique

line joining two distinct points. The above result can be stated in the form of an axiom as

follows.

Axiom: Given two distinct points, there is a unique line that passes through them.

Postulate 2: A terminated line can be produced indefinitely.

The second postulate states that a line segment can be extended on either side to form a

line.

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.

A system of axioms is called consistent, if it is impossible to deduce a statement from

these axioms that contradicts any axiom or previously proved statement.

Therefore, when a system of axioms is given, it has to be ensured that the system is

consistent.

Propositions or theorems

Propositions or theorems are statements that are proved, using definitions, axioms,

previously proved statements, and deductive reasoning.

Theorem: Two distinct lines cannot have more than one point in common.

This theorem can be proved by using the axiom, “There is a unique line passing through

two distinct points”.

Equivalent versions of Euclid’s fifth postulate

Two equivalent versions of Euclid‟s fifth postulate are as follows.

For every line l and for every point p not lying on l, there exists a unique line „m‟

passing through p and parallel to l.

Two distinct intersecting lines cannot be parallel to the same line.

The attempts to prove Euclid‟s fifth postulate as a theorem have failed. However, their

efforts have led to the discovery of several other geometries called non-Euclidean

geometries.

Non-Euclidean geometry is also called spherical geometry. In spherical geometry, lines

are not straight. They are part of great circles (that is, circles obtained by the intersection

of a sphere and planes passing through the centre of the sphere).

Chapter 6: Lines and Angles

A pair of angles whose sum is 90 is called complementary angles.

Example: 40 and 50 are complementary angles.

A pair of angles whose sum is 180 is known as supplementary angles.

Example: 60 and 120 are supplementary angles.

If two lines intersect each other

The pairs of opposite angles so formed are called pairs of vertically opposite angles.

Vertically opposite angles are equal in measure

Example: In the following figure, AOD and BOC, AOC and BOD are the pairs of

vertically opposite angles.

AOD = BOC and AOC = BOD

Two angles are said to be adjacent angles, if they have a common arm.

In the given figure, AOB and BOC are adjacent angles.

A pair of angles is called a linear pair, if they are adjacent and supplementary.

In the given figure, ABD and CBD are linear pair of angles.

It can be said that if a ray stands on a line, then the two angles so formed are a linear pair

of angles.

Transversal is a line which intersects two or more lines at distinct points.

When a transversal intersects two lines l and m, the angles so formed at the intersection

points are named as follows.

Corresponding angles

1 and 5, 2 and 6, 3 and 7, 4 and 8

Alternate interior angles

3 and 5, 4 and 6

Alternate exterior angles

1 and 7, 2 and 8

Corresponding angle axiom and its converse

If a transversal intersects two parallel lines, then each pair of corresponding angles is

equal.

Its converse is also true.

If a transversal intersects two lines such that a pair of corresponding angles is equal, then

the two lines are parallel to each other.

In the following figure, the corresponding angles are equal. Therefore, the lines l and m

are parallel to each other.

Alternate angle axiom and its converse

If a transversal intersects two parallel lines, then each pair of alternate angles is equal.

Its converse is also true.

If a transversal intersects two lines such that a pair of alternate angles is equal, then the

two lines are parallel to each other.

In the following figure, a pair of alternate angles is equal. Therefore, l and m are parallel

lines.

Angles on the same side of transversal

If a transversal intersects two parallel lines, then each pair of angles on the same side of

the transversal are supplementary.

Its converse states that if a transversal intersects two lines such that each pair of interior

angles on the same side of the transversal are supplementary, then the two lines are

parallel to each other.

In the following figure, if 1 + 4 = 180 or 2 + 3 = 180, then it can be said that

lines l and m are parallel to each other.

Lines that are parallel to the same line are parallel to each other.

In the following figure, if AB||CD and CD||EF, then AB||EF.

Lines that are perpendicular to the same line are parallel to each other.

In the following figure, CEAB and DFAB. Hence, CE||DF.

Angle sum property

The sum of all the three interior angles of a triangle is 180.

A + B + C = 180

Exterior angle property

If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of

the two interior opposite angles.

ACX = BAC + ABC.

Chapter 7: Triangles

Two figures are said to be congruent if they are of the same shape and size.

Similar figures are of the same shape but not necessarily of the same size.

If ABC XYZ, then

AB = XY, BC = YZ, AC = XZ

A = X, B = Y, and C = Z.

Corresponding parts of congruent triangles are equal.

SAS congruence rule

If two sides and the included angle of one triangle are equal to the two sides and the

included angle of the other triangle, then the two triangles are congruent to each other.

ASA congruence rule

If two angles and the included side of a triangle are equal to the two angles and the

included side of the other triangle, then the two triangles are congruent to each other.

AAS congruence rule

If two angles and one side of a triangle are equal to two angles and the corresponding side

of the other triangle, then the two triangles are congruent to each other.

SSS congruence rule

If three sides of a triangle are equal to the three sides of the other triangle, then the two

triangles are congruent.

RHS congruence rule

If the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side

of the other right triangle, then the two triangles are congruent to each other.

Properties of isosceles triangles

Angles opposite to equal sides of a triangle are equal.

Sides opposite to equal angles of a triangle are equal in length.

Inequalities in a triangle

Angle opposite to the longer side of a triangle is greater.

Side opposite to the greater angle of a triangle is longer.

The sum of any two sides of a triangle is greater than the third side.

The difference of any two sides of a triangle is smaller than the third side.

Chapter 12: Heron’s Formula

Heron’s formula

When all the three sides of a triangle are given, its area can be calculated by Heron‟s

formula.

Let a, b, and c be the sides of a triangle.

Semi-perimeter of the triangle and is given by, 2

a b cs

Area of triangle = s s a s b s c

Example: What is the area of a triangle whose sides are 9 cm, 28 cm, and 35 cm?

Solution: Let a = 9 cm, b = 28 cm, and c = 35 cm

Semi-perimeter, 9 28 35

cm 36 cm2 2

a b cs

Area of triangle 36 36 9 36 28 6 35 cm2

2

2

36 27 8 1 cm

36 6 cm

Area of a quadrilateral can also be calculated using Heron‟s formula. Firstly, the

quadrilateral is divided into two triangles. Then, the area of each triangle is calculated

using Heron‟s formula.