**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 form $\frac{p}{q}$, where

*p*,

*q*are integers and

*q*≠ 0.

**Example**:

$\frac{1}{2},\frac{3}{4},\frac{6}{9},\mathrm{etc}.$

• Every rational number ‘

*x*’can be expressed as $x=\frac{a}{b}$, 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 $\frac{3}{8}\mathrm{and}\frac{5}{12}$.

**Solution:**

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

Now, $\frac{3}{8}+\frac{5}{12}=\frac{19}{24}$

∴ A rational number between $\frac{3}{8}\mathrm{and}\frac{5}{12}=\frac{\left({\displaystyle \frac{3}{8}}+{\displaystyle \frac{5}{12}}\right)}{2}=\frac{19}{48}$

❖

**Irrational numbers**

Irrational numbers are those which

**cannot**be expressed in the form$\frac{p}{q}$, where

*p*,

*q*are integers and

*q*≠ 0.

**Example**: $\mathrm{\pi},\sqrt{2},\sqrt{7},\sqrt{14},0.002202220...........$

• There are infinitely many irrational numbers.

• π = 3.141592… is irrational. Its approximate value is assumed as $\frac{22}{7}$ 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 $\sqrt{6}$ 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, $\mathrm{OB}=\sqrt{5}$

(b) Construct BD of unit length perpendicular to OB. Thus, by Pythagoras theorem, $\mathrm{OD}=\sqrt{\left({\sqrt{5}}^{2}\right)+12}=\sqrt{6}$

(c) With centre O and radius OD, draw an arc intersecting the number line at point P. Thus, P corresponds to the number $\sqrt{6}$.

❖

**Real numbers and their decimal expansions**

• The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating).

**Example:**

**$\frac{15}{8}=1.875\left(\mathrm{Terminating}\right)\phantom{\rule{0ex}{0ex}}\frac{4}{3}=1.333..........=1\overline{3}(\mathrm{Non}-\mathrm{terminating}\mathrm{recurring})$**

• 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:**$3.\overline{32}$ 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

• $\sqrt{ab}=\sqrt{a}\sqrt{b}$

• $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

• $\left(\sqrt{a}+b\right)\left(\sqrt{a}-b\right)=a-{b}^{2}$

• $\left(a+\sqrt{b}\right)\left(a-\sqrt{b}\right)={a}^{2}-b$

• $\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{c}+\sqrt{d}\right)=\sqrt{ac}+\sqrt{ad}+\sqrt{bc}+\sqrt{bd}$

• ${\left(\sqrt{a}+\sqrt{b}\right)}^{2}=a+2\sqrt{ab}+b$

❖

**Rationalisation of denominator**

The denominator of $\frac{\sqrt{a}+\sqrt{b}}{\sqrt{x}+\sqrt{y}}$can be rationalised by multiplying both the numerator and the denominator by $\sqrt{x}-\sqrt{y}$, where

*a*,

*b*,

*x*,

*y*are integers.

❖

**Laws of exponents**

Let

*a*> 0 is a real number and

*p*,

*q*are rational numbers.

• ${a}^{p}.{a}^{q}={a}^{p+q}$

• ${\left({a}^{p}\right)}^{q}={a}^{pq}$

• $\frac{{a}^{p}}{{a}^{q}}={a}^{p-q}$

• ${\left(ab\right)}^{p}={a}^{p}{b}^{p}$

• $\sqrt[x]{a}={a}^{\frac{1}{x}},\mathrm{when}n\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{integer}.$

**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 $p\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+.....+{a}_{1}x+{a}_{0}$, where ${a}_{0},{a}_{1}...{a}_{n}$ are constants and ${a}_{n}\ne 0$.

are the respective coefficients of and

*n*is called the degree of the polynomial. ${a}_{n}{x}^{n},{a}_{n-1}{x}^{n-1}...{a}_{0}\mathrm{and}{a}_{0}\ne 0$ 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., 3

*x*, 5, 25

*t*

^{3}.

• A polynomial comprising two terms is called a binomial, e.g., 2

*t*– 6, 3

*x*

^{4}+ 2

*x*etc.

• A polynomial comprising three terms is called a trinomial, e.g., $-{x}^{3}+\sqrt{5}x+2,{y}^{6}+y+9$.

❖

**Classification of polynomial according to their degrees**

• A polynomial of degree one is called a linear polynomial, e.g., 3

*x*+ 2, 4

*x*,

*x*+ 9.

• A polynomial of degree two is called a quadratic polynomial, e.g., ${x}^{2}+9,3{x}^{2}+4x+6.$

• A polynomial of degree three is called a cubic polynomial, e.g., 10

*x*

^{3}+ 3, 9

*x*

^{3}.

❖

**Zeroes of a polynomial:**A real number a is said to be the zero of polynomial

*p*(

*x*) if

*p*(

*α*) = 0. In this case, a is also called the root of the equation

*p*(

*x*) = 0

• 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 $p\left(x\right)=3{x}^{3}+2x+9\mathrm{at}x=-2$

**Solution:**

$p\left(x\right)=3{x}^{3}+2x+9\phantom{\rule{0ex}{0ex}}\therefore p\left(-2\right)=3{\left(-2\right)}^{3}+2\left(-2\right)+9\phantom{\rule{0ex}{0ex}}=-24-4+9=-19$

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

*g*(

*x*) ≠ 0, then we can find polynomials

*q*(

*x*) and

*r*(

*x*) such that

$p\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$, where

*r*(

*x*) = 0 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 ${x}^{4}-2{x}^{3}-2{x}^{2}+7x-15$ by

*x*– 2.

**Solution:**

It can be easily verified that $\left({x}^{4}-2{x}^{3}-2{x}^{2}+7x-15\right)=\left(x-2\right)\left({x}^{3}-2x+3\right)+\left(-9\right)$.

❖

**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*³ 1and

*a*is any real number, then

•

*x – a*is a factor of

*p*(

*x*), if

*p*(

*a*) = 0

•

*p*(

*a*)

*=*0

*,*if

*(*

*x – a*) is a factor of

*p*(

*x*)

❖

**Factorisation of polynomials:**Polynomials can be factorised by using the algebraic identities given below.

• ${\left(x+y\right)}^{2}={x}^{2}+2xy+{y}^{2}$

• ${\left(x-y\right)}^{2}={x}^{2}-2xy+{y}^{2}$

• $\left(x+y\right)\left(x-y\right)={x}^{2}-{y}^{2}$

• $\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+ab$

• ${\left(x+y+z\right)}^{2}={x}^{2}+{y}^{2}+{z}^{2}+2xy+2yz+2zx$

• ${\left(x+y\right)}^{3}={x}^{3}+{y}^{3}+3xy\left(x+y\right)={x}^{3}+{y}^{3}+3{x}^{2}y+3x{y}^{2}$

• ${\left(x-y\right)}^{3}={x}^{3}-{y}^{3}-3xy\left(x-y\right)={x}^{3}-{y}^{3}-3{x}^{2}y+3x{y}^{2}$

• ${x}^{3}+{y}^{3}+{z}^{3}-3xyz=\left(x+y+z\right)\left({x}^{2}+{y}^{2}+{z}^{2}-xy-yz-zx\right)$

**For example:**Factorise

$4{x}^{2}-20xy+25{y}^{2}\phantom{\rule{0ex}{0ex}}4{x}^{2}-20xy+25{y}^{2}={\left(2x\right)}^{2}-2\times 2x\times 5y+{\left(5y\right)}^{2}\phantom{\rule{0ex}{0ex}}={\left(2x-5y\right)}^{2}\left[{a}^{2}-2ab+{b}^{2}={\left(a-b\right)}^{2}\right]\phantom{\rule{0ex}{0ex}}=\left(2x-5y\right)\left(2x-5y\right)$

**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 1

^{st}quadrant is enclosed by the positive

*x*-axis and positive

*y*-axis. So, a point in the 1

^{st}quadrant is in the form (+, +).

• The 2

^{nd}quadrant is enclosed by the negative

*x*-axis and positive

*y*-axis. So, a point in the 2

^{nd}quadrant is in the form (–, +).

• The 3

^{rd}quadrant is enclosed by the negative

*x*-axis and the negative

*y*-axis. So, the point in the 3

^{rd}quadrant is in the form (–, –).

• The 4

^{th}quadrant is enclosed by the positive

*x*-axis and the negative

*y*-axis. So, the point in the 4

^{th}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, CE⊥AB and DF⊥AB. 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 ΔABC ≅ ΔXYZ, 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, $s=\frac{a+b+c}{2}$

• Area of triangle $=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

**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, $s=\frac{a+b+c}{2}=\left(\frac{9+28+35}{2}\right)\mathrm{cm}=36\mathrm{cm}$

$\mathrm{Area}\mathrm{of}triangle=\sqrt{36\left(36-9\right)\left(36-28\right)\left(6-35\right)}{\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}=\sqrt{36\times 27\times 8\times 1}{\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}=36\sqrt{6}{\mathrm{cm}}^{2}$

**❖**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.

**Chapter 4: Linear Equations in Two Variables**

❖ An equation of the form,

*ax + by + c*= 0, where

*a*,

*b*, and

*c*are real numbers, such that

*a*and

*b*are both not zero, is called a linear equation in two variables.

For example, 2

*x*+ 3

*y*+ 10 = 0, 3

*x*+ 7

*y*= 0

• Equations of the form,

*ax + b*= 0 or

*cy*+

*d*= 0, are also examples of linear equations in two variables since they can be written as

*ax +*0.

*y + b*= 0 or 0.

*x*+

*cy*+

*d*= 0 respectively.

❖

**Solution of a linear equation in two variables**

A solution of a linear equation represents a pair of values, one for

*x*and one for

*y*which satisfy the given equation.

• Linear equation in one variable has a unique solution.

• Linear equation in two variables has infinitely many solutions.

❖

**Geometrical representation of a linear equation in two variables**

• The geometrical representation of the equation,

*ax + by + c*= 0, is a straight line.

**For example:**the equation

*x*+ 3

*y*= 6 can be represented on a graph paper as follows:

• An equation of the form,

*y*=

*mx*, represents a line passing through the origin.

• Every point on the graph of a linear equation in two variables is a solution of the linear equation. Every solution of the linear equation is a point on the graph of the linear equation.

❖

**Equation of lines parallel to the**

*x*-axis and*y*-axis• The graph of

*x*=

*a*is a straight line parallel to the

*y*-axis.

• The graph of

*y*=

*b*is a straight line parallel to the

*x*-axis.

**Chapter 8: Quadrilaterals**

❖ The sum of all the interior angles of a quadrilateral is 360°.

In the following quadrilateral ABCD, ∠A + ∠B + ∠C + ∠D = 360°.

❖

**Classification of quadrilaterals**

❖

**Classification of parallelograms**

❖ Diagonals of a parallelogram divide it into two congruent triangles.

If ABCD is a parallelogram, then ΔABC ≅ ΔCDA

❖ In a parallelogram,

• opposite sides are parallel and equal

• opposite angles are equal

• adjacent angles are supplementary

• diagonals bisect each other

❖ A quadrilateral is a parallelogram, if

• each pair of opposite sides is equal

• each pair of opposite angles is equal

• diagonals bisect each other

• a pair of opposite sides is equal and parallel

❖

**Properties of some special parallelograms**

• Diagonals of a rectangle are equal and bisect each other.

• Diagonals of a rhombus bisect each other at right angles.

• Diagonals of a square are equal and bisect each other at right angles.

❖

**Mid-point theorem and its converse**

•

**Mid-point theorem**

The line segment joining the mid-point of any two sides of a triangle is parallel to the third side and is half of it.

In ΔABC, if D and E are the mid-points of sides AB and AC respectively, then by the mid-point theorem, DE||BC and DE =$\frac{\mathrm{BC}}{2}$.

•

**Converse of the mid-point theorem**

A line through the mid-point of one side of a triangle parallel to the other side bisects the third side.

In the given figure, if AP = PB and PQ||BC, then PQ bisects AB i.e., Q is the mid-point of AC.

❖ The quadrilateral formed by joining the mid-points of the sides of a quadrilateral is a parallelogram.

**Chapter 9: Areas of Parallelograms and Triangles**

❖ Two congruent figures have equal areas, but the converse is not true.

❖ If a figure is formed by two non-overlapping regions A and B, then the area of the figure = Area (A) + Area (B).

❖ Two figures lie on the same base and between the same parallels if they have a common base and if the opposite vertex (or side) lies on a line parallel to the base.

**For example:**

In the first figure, parallelograms ABCE and ABDF lie on the same base AB and between the same parallels AB and CF.

In the second figure, parallelogram ABCE and triangle ABD lie on the same base but they do not lie between the same parallels.

❖ Parallelograms on the same (or equal) base and between the same parallels are equal in area.

Its converse is also true, i.e., parallelograms on the same base and having equal areas lie between the same parallels.

❖ The area of a parallelogram is the product of its base and the corresponding height.

❖

**Areas of parallelogram and triangle on the same base**

• If a parallelogram and a triangle lie on the same (or equal) base and between the same parallels, then the area of the triangle is half the area of the parallelogram.

• If a parallelogram and a triangle lie on the same base and the area of the triangle is half the area of the parallelogram, then the triangle and the parallelogram lie between the same parallels.

❖

**Triangles on the same base and between the same parallels**

• Triangles on the same base (or equal base) and between the same parallels are equal in area.

• Triangles having the same base and equal areas lie between the same parallels.

❖ The median of a triangle divides it into two congruent triangles.

Thus, a median of a triangle divides it into two triangles of equal area.

**Chapter 10: Circles**

❖ Two or more circles are said to be congruent if they have the same radii.

❖ Two or more circles are said to be concentric if their centre’s lie at the same point.

❖

**Angle subtended by chords at the centre**

• Chords that are equal in length subtend equal angles at the centre of the circle.

• Chords subtending equal angles at the centre of the circle are equal in length.

❖

**Perpendicular from the centre to a chord**

• The perpendicular from the centre of a circle to a chord bisects the chord.

• The line joining the centre of the circle to the mid-point of a chord is perpendicular to the chord.

• Perpendicular bisector of a chord always passes through the centre of the circle.

❖ At least three points are required to construct a unique circle.

❖

**Equal chords and their distances from the centre**

• Equal chords of a circle (or congruent circles) are equidistant from the centre.

• Chords which are equidistant from the centre of a circle are equal in length.

❖

**Angle subtended by an arc of a circle**

• Two or more chords are equal if and only if the corresponding arcs are congruent.

• The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at the remaining part of the circle.

For example:

Here, ∠AXB is the angle subtended by arc AB at the remaining part of the circle.

∴ ∠AOB = 2 × ∠AXB

$\Rightarrow \angle \mathrm{AXB}=\frac{1}{2}\times 30\xb0=15\xb0$

• Angles in the same segment of a circle are equal.

∠PRQ and ∠PSQ lie in the same segment of a circle.

∴ ∠PRQ = ∠PSQ

• Angle in a semicircle is a right angle.

❖

**Concyclic points**

• A set of points that lie on a common circle are known as concyclic points.

Here, A, B, D, and E are concyclic points.

• If a line segment joining two points subtends equal angles at the two points lying on the same side of the line segment, then the four points are concyclic.

For example:

Here, if ∠ACB = ∠ADB, then points A, B, C, and D are concyclic.

❖

**Cyclic quadrilaterals**

• A cyclic quadrilateral is a quadrilateral if all four vertices of the quadrilateral lie on a circle.

Here, ABCD is a cyclic quadrilateral.

• The sum of each pair of opposite angles of a cyclic quadrilateral is 180°.

• If the sum of a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.

❖ The quadrilateral formed by the angle bisectors of interior angles of any quadrilateral is a cyclic quadrilateral.

❖ Non-parallel sides of a cyclic trapezium are equal in length.

**Chapter 11: Constructions**

❖

**When the base, base angle, and the sum of other two sides of a triangle are given, the triangle can be constructed as follows.**

Let us suppose base BC, ∠B, and (AB + AC) are given.

**Steps of construction:**

(1) Draw BC and make an angle, ∠B, at point B.

(2) Draw an arc on BX, which cuts it at point P, such that BP (= AB + AC).

(3) Join PC and draw its perpendicular bisector. Let this perpendicular bisector intersect BP at A.

Thus, ∆ABC is the required triangle.

❖

**When the base, base angle, and the difference between the other two sides of the triangle are given, the triangle can be constructed as follows.**

Let us suppose base BC, ∠B, and (AB – AC) are given.

**Steps of construction:**

(1) Draw BC and make an angle, ∠B, at point B.

(2) Draw an arc on BX, which cuts it at point P, such that BP (= AB – AC).

(3) Join PC and draw its perpendicular bisector. Let this perpendicular bisector intersect BX at point A. Join AC.

Thus, ∆ABC is the required triangle.

❖

**When the base BC,**

**∠**

**B, and (AC – AB) are given, the triangle can be constructed as follows**:

(1) Draw base BC and ∠B.

(2) Draw an arc, which cuts extended BX on opposite side of BC at point Q, such that

BQ = (AC – AB).

(3) Join QC and draw its perpendicular bisector. Let this perpendicular bisector intersect BX at point A. Join AC.

Thus, ∆ABC is the required triangle.

❖

**When the perimeter and two base angles of the triangle are given, the triangle can be constructed as follows.**

Let us suppose that base angle, ∠B, and ∠C of ∆ABC are given.

**Steps of construction:**

(1) Draw a line segment PQ of length equal to the perimeter of the triangle and draw the base angles at points P and Q.

(2) Draw the angle bisectors of ∠P and ∠Q. Let these angle bisectors intersect each other at point A.

(3) Draw the perpendicular bisectors of AP and AQ. Let these perpendicular bisectors

intersect PQ at points B and C respectively. Join AB and AC.

Thus, ∆ABC is the required triangle.

**Chapter 13: Surface Areas and Volumes**

**❖ Cuboid**

Consider a cuboid with dimensions

*l*,

*b*and

*h*

• Lateral surface area = 2

*h*(

*l*+

*b*)

• Total surface area = 2(

*lb*+

*bh*+

*hl*)

• Volume =

*l × b × h*

**❖ Cube**

Consider a cube with edge length

*a*

• Lateral surface area = 4

*a*

^{2}

• Total surface area = 6

*a*

^{2}

• Volume =

*a*

^{3}

**❖ Right circular cylinder**

Consider a right circular cylinder with height

*h*and radius

*r*.

• Curved surface area = 2π

*rh*

•Total surface area

•

*= 2π*

*r*(*h + r*)• Volume = π

*r*

^{2}

*h*

**❖ Right circular cone**

Consider a right circular cone of height

*h*, slant height

*l*and base radius

*r*

• Slant height,

*$l=\sqrt{{h}^{2}+{r}^{2}}$*

• Curved surface area = π

*rl*

• Total surface area = π

*r*(

*l + r*)

• Volume $\frac{1}{3}\pi {r}^{2}h$

**❖ Sphere**

Consider a sphere of radius

*r*.

• Curved surface area = Total surface area = 4π

*r*

^{2}• Volume $=\frac{4}{3}\pi {r}^{3}$

**❖ Hemisphere**

Consider a hemisphere of radius

*r*.

• Curved surface area = 2π

*r*

^{2}

• Total surface area = 3π

*r*

^{2}

• Volume $=\frac{2}{3}\pi {r}^{3}$

**Chapter 14: Statistics**

❖ The marks of 20 students of a school are as follows.

86 | 49 | 52 | 78 | 46 | 54 |

62 | 71 | 92 | 87 | 84 | 45 |

58 | 52 | 50 | 60 | 77 | 85 |

88 | 63 |

The above data can be written in the form of class intervals as follows.

Marks |
Number of students |

40 – 50 | 3 |

50 – 60 | 5 |

60 –70 | 3 |

70 – 80 | 3 |

80 –90 | 5 |

90 –100 | 1 |

• This table is called grouped frequency distribution table.

• 40 – 50, 50 – 60, etc. are class intervals.

• 40 is the lower limit and 50 is the upper limit of class interval 40 – 50.

• The number of students for each class interval is the frequency of that class interval.

❖

**Exclusive frequency distribution table**

The frequency distribution tables in which the upper limit of any class interval coincides with the lower limit of the next class interval are known as exclusive frequency distribution tables.

❖

**Inclusive frequency distribution table**

Consider the following table.

Class interval |
Frequency |

10 – 19 | 2 |

20 – 29 | 7 |

30 – 39 | 4 |

40 – 49 | 1 |

Here, the upper limit of any class interval does not coincide with the lower class limit of next class interval. Such frequency distribution table is known as inclusive frequency distribution table.

It can be converted into exclusive table by subtracting $\frac{20-19}{2}=\frac{1}{2}=0.5$ from the upper limit and lower limit of each class interval as follows.

Class interval |
Class interval |
Frequency |

(10 – 0.5) – (19 + 0.5) | 9.5 –19.5 | 2 |

(20 – 0.5) – (29 + 0.5) | 19.5 – 29.5 | 7 |

(30 – 0.5) – (39 + 0.5) | 29.5 –39.5 | 4 |

(40 – 0.5) – (49 + 0.5) | 39.5 – 49.5 | 1 |

❖ A histogram is a graphical representation of data.

**Example:**Represent the given data in the form of a histogram.

Height (in cm) |
Number of students |

140 – 150 | 10 |

150 – 160 | 6 |

160 – 170 | 15 |

170 –180 | 4 |

**Solution:**

For this data, histogram can be drawn by taking class intervals along

*x*-axis and frequency along

*y*-axis and then drawing bars parallel to

*y*-axis.

The histogram for this data is as follows.

❖ If the class intervals are not of uniform width, then in the histogram, the length of bars is equal to adjusted frequencies. For example, consider the following data.

Class interval |
Frequency |

0 – 10 | 1 |

10 – 20 | 5 |

20 – 40 | 8 |

40 –70 | 6 |

Here, minimum class size is 10.

The adjusted frequencies can be calculated by, $\frac{\mathrm{Frequency}}{\mathrm{Class}\mathrm{width}}$× Minimum class size

Therefore, we obtain the table as:

Class interval |
Frequency |
Adjusted frequency |

0 – 10 | 1 | $\frac{1}{10}\times 10=1$ |

10 – 20 | 5 | $\frac{5}{10}\times 10=5$ |

20 – 40 | 8 | $\frac{8}{20}\times 10=4$ |

40 – 70 | 6 | $\frac{6}{30}\times 10=2$ |

The histogram can be drawn by taking class intervals on

*x*-axis and adjusted frequencies on

*y*-axis.

❖ The frequency polygon for a grouped data is drawn by first drawing its histogram and then by joining the mid-points of the top of bars. For example, the frequency polygon for the data given in the previous table can be drawn as follows.

ABCDEF is the required frequency polygon.

**❖ Measures of central tendency**

Mean, median, and mode are the measures of central tendency.

**•**

**Mean:**Mean is defined by,

$\mathrm{Mean}\left(\overline{\mathit{x}}\right)=\frac{\mathrm{Sum}\mathrm{of}\mathrm{all}\mathrm{observations}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{observations}}=\frac{\sum {f}_{\mathit{i}}{x}_{\mathit{i}}}{\sum {f}_{\mathit{i}}}$

•

**Median:**To find the median, the observations are arranged in ascending or descending order.

(i) If the number of observations (

*n*) is odd, then value of $\frac{{\left(n+1\right)}^{\mathrm{th}}}{2}$ observation is the median.

(ii) If the number of observations (

*n*) is even, then the mean of the values of $\frac{{\left(n\right)}^{\mathrm{th}}}{2}\mathrm{and}{\left(\frac{n}{2}+1\right)}^{\mathrm{th}}$ observations is the median.

•

**Mode:**The value of the observation that occurs most frequently is called mode. Mode is the value of observation whose frequency is maximum.

**Chapter 15: Probability**

❖

**Experiment:**An experiment is a situation involving chance or probability.

For example, tossing a coin is an experiment.

❖

**Outcome:**An outcome is the result of an experiment.

For example, getting a head on tossing a coin is an outcome.

❖

**Sample space:**The set of all possible outcomes of an experiment is called sample space.

The sample space of the experiment of throwing a die is {1, 2, 3, 4, 5, 6}.

❖

**Event:**An event is the set of one or more outcomes of an experiment.

For example, in the experiment of throwing a die, the event of getting an even number is {2, 4, 6}.

❖

**Probability:**The empirical (or experimental) probability of an event

*A*is given by

$P\left(A\right)=\frac{\mathrm{Number}\mathrm{of}\mathrm{favourable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$

**Example:**When a coin is tossed 500 times and on the upper face of the coin tail comes up 280 times, what is the probability of getting head on the upper face of the coin?

**Solution:**Let A be the event of getting head on the upper face of the coin.

Total number of trials = 500

Number of trials in which tail comes up = 280

∴ Number of trials in which head comes up = 500 – 280 = 220

$\therefore \mathrm{P}\left(\mathrm{A}\right)=\frac{220}{500}=\frac{11}{25}$

❖The probability of an event always lies between 0 and 1.