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Operations on Integers
The set of natural numbers, zero and the negatives of natural numbers form the set of Integers.
The set of integers includes all the whole numbers. There is no smallest integer.
Addition of Integers
The sum of two positive integers results in a positive integer.
Ex: 8 + 2 = 10
The sum of two negative integers results in a negative integer.
Ex: (– 6) + (– 3) = – 9
The sum of a positive and a negative integer is the difference of the numbers with the sign of the
larger integer of the two.
Ex: 45 + (– 25) = 20 and (– 45) + 20 = – 25
The additive inverse of any integer 'a' is '– a', and the additive inverse of '– a' is 'a'.
Ex: Additive inverse of (– 12) = – (– 12) = 12
Subtraction of Integers
Subtraction is the opposite of addition, Therefore, to subtract two integers, we add the additive
inverse of the integer that is being subtracted to the other integer.
Ex: 23 – 43 = 23 + (Additive inverse of 43) = 23 + (– 43) = – 20
Multiplication of Integers
The product of two positive integers is a positive integer. The product of a positive and a
negative integer is a negative integer. The product of two negative integers is a positive integer.
If the number of negative integers in a product is even, then the product is a positive integer.
Similarly, if the number of negative integers in a product is odd, then the product is a negative
integer.
Division of Integers
Division is the inverse operation of multiplication. The division of a positive integer by a
positive integer results in a positive integer. The division of a negative integer by a positive
integer results in a negative integer. The division of a positive integer by a negative integer
results in a negative integer. The division of a negative integer by a negative integer results in a
positive integer.
For any integer 'a',
a × 0 = 0 × a = 0.
a ÷ 0 is not defined.
0 ÷ a = 0, where a is not equal to zero.
Properties of Integers
Closure property
Closure property under addition:
Integers are closed under addition, i.e. for any two integers a and b, a + b is an integer.
Ex: 3 + 4 = 7; (– 9) + 7 = – 2.
Closure property under subtraction:
Integers are closed under subtraction, i.e. for any two integers a and b, a – b is an integer.
Ex: (– 21) – (– 9) = (– 12); 8 – 3 = 5.
Closure property under multiplication:
Integers are closed under multiplication, i.e. for any two integers a and b, ab is an integer.
Ex: 5 × 6 = 30; (– 9) × (– 3) = 27.
Closure property under division:
Integers are not closed under division, i.e. for any two integers a and b, ababmay not be an
integer.
Ex:(– 2) ÷ (– 4) = 1212
Commutative property
Commutative property under addition:
Addition is commutative for integers. For any two integers a and b, a + b = b + a.
Ex: 5 + (– 6) = 5 – 6 = – 1;
(– 6) + 5 = – 6 + 5 = –1
∴ 5 + (– 6) = (– 6) + 5.
Commutative property under subtraction:
Subtraction is not commutative for integers. For any two integers a and b, a – b ≠ b – a.
Ex: 8 – (– 6) = 8 + 6 = 14;
(– 6) – 8 = – 6 – 8 = – 14
∴ 8 – (– 6) ≠ – 6 – 8.
Commutative property under multiplication:
Multiplication is commutative for integers. For any two integers a and b, ab = ba.
Ex: 9 × (– 6) = – (9 × 6) = – 54;
(– 6) × 9 = – (6 × 9) = – 54
∴ 9 × (– 6) = (– 6) × 9.
Commutative property under division:
Division is not commutative for integers. For any two integers a and b, a ÷ b ≠ b ÷ a.
Ex: (– 14) ÷ 2 = – 7
2 ÷ (–14) = – 1717
(– 14) ÷ 2 ≠ 2 ÷ (–14).
Associative property
Associative property under addition:
Addition is associative for integers. For any three integers a, b and c, a + (b + c) = (a + b) + c
Ex: 5 + (– 6 + 4) = 5 + (– 2) = 3;
(5 – 6) + 4 = (– 1) + 4 = 3
∴ 5 + (– 6 + 4) = (5 – 6) + 4.
Associative property under subtraction:
Subtraction is associative for integers. For any three integers a, b and c, a – (b – c) ≠ (a – b) – c
Ex: 5 – (6 – 4) = 5 – 2 = 3;
(5 – 6) – 4 = – 1 – 4 = – 5
∴ 5 – (6 – 4) ≠ (5 – 6) – 4.
Associative property under multiplication:
Multiplication is associative for integers. For any three integers a, b and c, (a × b) × c = a × (b ×
c)
Ex: [(– 3) × (– 2)] × 4 = (6 × 4) = 24
(– 3) × [(– 2) × 4] = (– 3) × (– 8) = 24
∴ [(– 3) × (– 2)] × 4 = [(– 3) × (– 2) × 4].
Associative property under division:
Division is not associative for integers.
Distributive property
Distributive property of multiplication over addition:
For any three integers a, b and c, a × (b + c) = (a × b) + (a × c).
Ex: – 2 (4 + 3) = –2 (7) = –14
= (– 2 × 4) + (– 2 × 3)
= (– 8) + (– 6)
= – 14.
Distributive property of multiplication over subtraction:
For any three integers, a, b and c, a × (b - c) = (a × b) – (a × c).
Ex: – 2 (4 – 3) = – 2 (1) = – 2
= (–2 × 4) – (– 2 × 3)
= (– 8) – (– 6)
= – 2.
The distributive property of multiplication over the operations of addition and subtraction is true
in the case of integers.
Identity under addition:
Integer 0 is the identity under addition. That is, for any integer a, a + 0 = 0 + a = a.
Ex: 4 + 0 = 0 + 4 = 4.
Identity under multiplication:
The integer 1 is the identity under multiplication. That is, for any integer a, 1 × a = a × 1 = a.
Ex: (– 4) × 1 = 1 × (– 4) = – 4.
When an integer is multiplied by –1, the result is the integer with sign changed i.e. the additive
identity of the integer.
For any integer a, a × –1 = –1 × a = –a.
Multiplication and Division of Fraction
Types of fractions
Proper fraction
A fraction whose numerator is less than the denominator is called the proper fraction. A proper
fraction is a fraction that represents a part of a whole.
Improper fraction
A fraction whose numerator is greater than the denominator is called an improper fraction. An
improper fraction is a combination of a whole and a proper fraction.
Mixed fraction
An improper fraction can be expressed as a mixed fraction. The numerator of the improper
fraction is divided by the denominator to obtain the quotient and the remainder. The mixed
fraction is written as Quotient RemainderDivisorQuotient RemainderDivisor.
Multiplication of fractions
To multiply a fraction by a whole number, multiply the whole number by the numerator of the
fraction.
Ex: 2 × 2323 = 4343.
While multiplying a whole number by a mixed fraction, change the mixed fraction into an
improper fraction and multiply the whole number by the numerator of the fraction.
Ex: 3 × 2 3434 = 3 × 114114 = 334334.
To multiply two fractions, multiply their numerators and denominators.
Ex: 3434 × 2525 = 3 × 24 × 53 × 24 × 5 = 620620.
When two proper fractions are multiplied, the product is less than each of the individual
fractions.
When two improper fractions are multiplied, the product is greater than each of the individual
fractions.
Fraction as an operator ‘of ’
Fraction acts as an operator of. 'of' represents multiplication.
Ex: 1313 of 90 = 1313 × 90 = 30.
Reciprocal of a fraction
To obtain the reciprocal of a fraction, interchange the numerator with the denominator.
Ex: The reciprocal of 3737 is 7373.
Division of fractions
To divide a whole number by a fraction, find the reciprocal of the fraction and then multiply it
by the whole number.
Ex: 2 ÷ 3434 = 2 × 4343 = 8383.
To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole
number.
Ex: 2727 ÷ 6 = 2727 × 1616 = 121121.
To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second
fraction.
Ex: 1212 ÷ 3434 = 1212 × 4343 = 4646.
To divide a whole number by a mixed fraction, convert the mixed fraction into an improper
fraction and multiply the whole number by the reciprocal of the improper fraction.
Ex: 5 ÷ 61414 = 5 ÷ 254254 = 5 × 425425 = 4545.
Multiplication and Division on Decimals
Multiplication of decimals
Steps to multiply a whole number by a decimal number:
Step 1: Ignore the decimals and multiply the two numbers.
Step 2: Count the number of digits to the right of decimal point in the original decimal number.
Step 3: Insert the decimal in the answer by the same count from right to left.
e.g. (i) 3 × 0.2 = 0.6 (ii) 3 × 0.4 = 1.2
Steps to multiply a decimal number by a decimal number:
Step 1: Ignore the decimals and multiply the two numbers.
Step 2: Count the number of digits to the right of decimal point in both the decimal numbers
Step 3: Add the number of digits counted and insert the decimal in the answer by the same count
from right to left.
e.g. (i) 0.2 × 0.7 = 0.14 (ii) 0.9 × 0.02 = 0.018
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Multiplication of a decimal number by 10, 100 or 1000:
For multiplying a decimal number by 10, retain the original number and shift the decimal to the
right by one place
For multiplying a decimal number by 100, retain the original number and shift the decimal to the
right by two places
For multiplying a decimal number by 1000, retain the original number and shift the decimal to
the right by three places
Division of decimals
Steps to divide a decimal number by a whole number:
Step 1: Convert the decimal number into a fraction.
Step 2: Find the reciprocal of the divisor.
Step 3: Multiply the fraction by the reciprocal.
e.g. 0.14 ÷ 3 = 1410014100 × 1313 = 1430014300
Steps to divide a decimal number by another decimal number:
Step 1: Convert both the decimal numbers into fractions.
Step 2: Find the reciprocal of the divisor.
Step 3: Multiply the fraction by the reciprocal.
e.g. 0.45 ÷ 0.15 = 4510045100 ÷ 1510015100= 4510045100× 1001510015= 3
Data Representation
Any information collected can be first arranged in a frequency distribution table, and this
information can be put as a visual representation in the form of pictographs or bar graphs. Data
collected is organised in a table for easy understanding and interpreting. Graphs are a visual
representation of organised data.
Bar graphs
A bar graph is the representation of data using rectangular bars of uniform width, and with their
lengths depending on the frequency and the scale chosen. The bars can be plotted vertically or
horizontally. You can look at a bar graph and make deductions about the data.
Bar graphs are used for plotting discrete or discontinuous data, i.e. data that has discrete values
and is not continuous. Some examples of discontinuous data are 'shoe size' and 'eye colour', for
which you can use a bar chart. On the other hand, examples of continuous data include 'height'
and 'weight'.
A bar graph is very useful if you are trying to record certain information, whether the data is
continuous or not. Bar graphs can also be used for comparative analysis.
Double bar graphs are used for comparing data between two different things. The difference
between a bar graph and a double bar graph is that a bar graph displays one set of data, and a
double bar graph compares two different sets of information or data.
Scale of a bar graph
If numbers in units are to be shown in a bar graph, the graph represents one unit of ength for one
observation and if it has to show numbers in tens or hundreds, one unit length can represent 10 or
100 observations. This is called the scale of the graph. Length of a bar in a bar graph depends on
the frequency and the scale chosen.
Data Value
Different forms of representative or central value are used to represent and describe different
forms of data. One of the representative values is the “Arithmetic Mean”.
Arithmetic mean
Arithmetic mean is a number that lies between the highest and the lowest values of the data.
Arithmetic mean or Average = Sum of observationsNumber of observationsSum of
observationsNumber of observations
Note that we need not arrange the data in ascending or descending order to calculate arithmetic
mean.
Range
Range is calculated by subtracting the lowest observation from the highest observation of a data.
It gives an idea of the spread of the observations.
Range = Highest observation – Lowest observation
Mode
Mode of a set of observations refers to the observation that occurs most often.
Steps to calculate the mode of a data:
Step 1: Arrange the data in ascending order.
Step 2: Tabulate the data in a frequency distribution table.
Step 3: The observation with a greatest frequency is the mode.
Median
Median refers to the value that lies in the middle of the data with half of the observations above
it and the other half of the observations below it.
Steps to calculate the median of a data:
Step 1: Arrange the data in the ascending order.
Step 2: The value that lies in the middle such that half of the observations lie above it and the
other half below it will be the median.
The mean, mode and median are the representative values of a group of observations or data, and
lie between the minimum and maximum values of the data. They are also called the measures of
the central tendency.
Introduction to Simple Equations
Variable
A variable is something that varies. A variable can take different numerical values; its value is
not fixed. Variables are usually denoted by letters of the alphabet, such as x, y, z, l, m, n, p, q etc.
Expressions are formed using variables.
Expression
Expressions are formed by performing the operations like addition, subtraction, multiplication
and division on the variables.Value of an expression depends on the value of the variable using
which the expression is formed.
Equation
An equation is a condition of equality between two mathematical expressions. It is a condition on
a variable.
e.g. 2x - 3 = 5, 3x + 9 = 11, 4y + 2 = 12
Two expressions on both sides of the equation should have equal value and atleast one of the
expressions must have a variable. An equation remains the same if the LHS and the RHS of the
equation are interchanged.
The equality sign in an equation shows that the value of the expression to the left hand side is
equal to the value of the expression to the right hand side. If there is some other sign other than
the equality sign between the LHS and the RHS, it is not an equation.
e.g. 3x + 5 > 6 is not an equation.
Solution of an equation
The value of the variable for which the left hand side of an equation is equal to its right hand side
is called the solution of that equation.
e.g. For the equation, 5x + 5 = 15, x = 2 is a solution.
When the same number is added to or subtracted from both the sides of a balanced equation, the
value of the left hand side remains equal to its value on the right hand side. If the same
mathematical operation is not done on both sides of a balanced equation, the balance is
disturbed.
e.g. (1) 5x + 3 = 13
On adding 2 to both sides of the equation, we get
5x + 3 + 2 = 13 + 2
5x + 5 = 15
(2) On subtracting 2 from both sides of the equation, we get
5x + 3 - 2 = 13 - 2
5x + 1 = 11
When an equation is divided or multiplied on both the sides by a non-zero number, the value of
the left hand side remains equal to its value on the right hand side.
e.g. (1) 5x + 3 = 13
On dividing both sides of the equation by 4, we get
(5x + 3) ÷ 4 = 13 ÷ 4
2) 5x + 1 = 13
On multiplying both sides of the equation by 4, we get
4(5x + 1) = 4(13)
20x + 4 = 52
Application of Simple Equations
Solving an equation
To find the solution of an equation, a series of identical mathematical operations are performed
on both the sides of the equation so that only the variable remains on one side. On simplifying all
the numbers, the result obtained is the solution of the equation. If the operation is performed on
only one side of the equation, the balance of the equation is disturbed.
Ex: 3x + 8 = 83
3x + 8 - 8 = 83 - 8
3x = 75
x = 753753
x = 25.
Transposition
Moving a term of an equation from one side to the other side is called transposing. Transposing a
number is same as adding to or subtracting the same number from both sides of the equation.
Ex: Solve 2x + 8 = 24
Given, 2x + 8 = 24
Transposing 8 to the right hand side, we get
⇒ 2x = 24 - 8
⇒ 2x = 16
⇒ x = 162162
⇒ x = 8.
Hence, the value of x is 8.
The sign of a number changes when it is transposed from one side of the equation to the other.
To solve puzzles/problems from practical situations equations are formed corresponding to such
situations and then those equations are solved to give the solution to the puzzles/problems.
Solution to an equation
An equation can be built from the solution of the equation using the property of doing the same
mathematical operation on both sides of an equation.
Angles
Angle
An angle is formed when lines, rays or line segments meet. The lines that form an angle are
called the sides or the arms of the angle. The common end point is called the vertex of the angle.
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Types of angles based on their measure
The angle whose measure is less than 90° is called an acute angle.
The angle whose measure is 90° is called a right angle.
The angle whose measure is greater than 90° and less than 180° is called an obtuse angle.
The angle whose measure is equal to 180° is called a straight angle.
The angle whose measure is greater than 180° and less than 360° is called a reflex angle.
Related angles
Two angles sum of whose measures is 90° are called complementary angles. When two angles
are complementary, each angle is said to be the complement of the other.
Two angles sum of whose measures is 180° are called supplementary angles. When two angles
are supplementary, each angle is said to be the supplement of the other.
Angles with a common vertex and a common arm are called adjacent angles. Non–common arms
of adjacent angles are on either side of the common arm. Adjacent angles have no common
interior points. Two adjacent angles can be either supplementary or complementary.
A pair of adjacent angles whose non–common arms are opposite rays is called a linear pair. The
angles of a linear pair are supplementary.
Two angles that are not adjacent but have a common vertex are called vertically opposite angles.
Vertically opposite angles are opposite to each other and are equal. Two pairs of vertically
opposite angles are formed when two lines intersect. If one pair of vertically opposite angles are
acute angles, then the other pair of angles are obtuse.
Pairs of Lines
Intersecting lines
Lines that meet at a point are called intersecting lines. The point where they meet is known as the
point of intersection.
Parallel lines
Lines that always remain the same distance apart and never meet are called parallel lines.
Transversal
A line that intersects two or more lines at distinct points is called a transversal.
Angles formed by lines intersected by a transversal
• Eight angles are formed when a transversal intersects two lines. The angles that lie between
the lines are called interior angles. The angles that lie on the outer sides of the lines are called
exterior angles.
• Angles formed on the same side of a transversal, on the same side of the two lines and at
corresponding vertices are called corresponding angles.
• Angles formed on the opposite sides of the transversal at the two distinct points of
intersection and between the two lines are called alternate interior angles.
• Angles formed on the opposite sides of the transversal at the two distinct points of
intersection but outside the two lines are called alternate exterior angles.
• Angles that have different vertices, lie on the same side of the transversal and are interior
angles are called consecutive interior angles or allied or co-interior angles.
If two parallel lines are cut by a transversal then each pair of:
• Corresponding angles is equal in measure
• Alternate interior angles is equal in measure
• Interior angles on the same side of the transversal are supplementary
Two lines are said to be parallel if each pair of:
• Corresponding angles are equal
• Alternate interior angles are equal
• Interior angles on the same side of the transversal are supplementary
Triangles
A triangle is a closed figure made of three line segments. Every triangle has three sides, three
angles, and three vertices. These are known as the parts of a triangle.
Classification of triangles
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The types of triangles classified by their sides are:
Equilateral triangle
Isosceles triangle
Scalene triangle
The types of triangles classified by their angles are:
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Right angled triangle
Obtuse angled triangle
Acute angled triangle
Equilateral triangle:
A triangle in which all the sides are equal is called an equilateral triangle. All the three angles of
an equilateral triangle are also equal, and each measures 60°.
Isosceles triangle:
A triangle in which any two sides are equal is called an isosceles triangle. In an isosceles
triangle, the angles opposite the equal sides are called the base angles, and they are equal.
Scalene triangle:
A triangle in which no two sides are equal is called a Scalene triangle.
Acute-angled triangle:
A triangle with all its angles less than 90° is known as an acute-angled triangle.
Obtuse-angled triangle:
A triangle with one of its angles more than 90° and less than 180° is known as an obtuse-angled
triangle.
Right-angled triangle:
A triangle with one of its angles equal to 90° is known as a right-angled triangle. The side
opposite the vertex with 90° angle is called the hypotenuse, and is the longest side of the triangle.
Median
The line segment drawn from a vertex of a triangle to the midpoint of the opposite side is called
a median of the triangle. Three medians can be drawn from each vertex of a triangle. Medians of
a triangle are concurrent. The point of concurrence is called the centroid of the traingle, and is
denoted by G.
The centroid and medians of a triangle always lie inside the triangle. The centroid of a triangle
divides the median in the ratio 2:1.
Altitude
The line segment drawn from the vertex of a triangle which is perpendicular to the opposite side
of that triangle is called an altitude of the triangle. A triangle has three altitudes. The altitudes of
a triangle are concurrent. The point of concurrence is called the orthocentre, and is denoted by O.
The altitude and orthocentre of a triangle need not lie inside the triangle.
Properties of Triangles
Angle sum property
The sum of the three angles of a triangle is 180°.
e.g. If A, B and C are the angles of a triangle ABC, then ∠A + ∠B + ∠C = 180°.
Proof:
Consider a triangle ABC.
Let line XY be parallel to side BC at A.
AB is a transversal that cuts the line XY and AB, at A and B, respectively.
As the alternate interior angles are equal, ∠1 = ∠4 and ∠2 = ∠5.
∠4, ∠3 and ∠5 form linear angles, and their sum is equal to 180°.
⇒ ∠4 + ∠3 + ∠5 = 180°
⇒ ∠1 + ∠2 + ∠3 = 180°
Hence, the sum of the three angles of a triangle is 180°.
Exterior angle property
An exterior angle of a triangle is equal to the sum of its opposite interior angles.
e.g. If ∠4 is an exterior angle of ΔABC, ∠1 and ∠2 are the interior opposite angles, then ∠4 =
∠1 + ∠2.
The sum of the lengths of any two sides of a triangle is greater than the third side.
In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other
two sides are called its legs.
Pythagorean theorem
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the
other two sides.
If b and c are legs and a is the hypotenuse of a right angled triangle then, a2 = b2 + c2.
Converse of Pythagorean theorem
If the sum of the squares on two sides of a triangle is equal to the square of the third side, then
the triangle must be a right-angled triangle.
Congruence of Plane Figures
If two objects are of exactly the same shape and size, they are said to be congruent. The relation
between two congruent objects being congruent is called congruence.
A plane figure is any shape that can be drawn in two dimensions e.g. rectangle, square, triangle,
rhombus, etc. To check if two figures drawn on a paper are congruent, make a traced copy of one
of the figures on a tracing paper and place it over the other. The other method is to cut out one of
these figures and place it over the
The method of superposition examines the congruence of plane figures, line segments and
angles. Two plane figures are congruent if each, when superimposed on the other, covers it
exactly. Congruence is denoted by ≅.
e.g. Two plane figures, say, P1 and P2, are congruent if the trace copy of P1 fits exactly on that of
P2. We write P1 ≅ P2
If two line segments have the same or equal length, they are congruent. Also, if two line
segments are congruent, then they have the same length.
e.g. Two line segments, say, AB−−−AB_ and EF−−−EF_ are congruent if they have equal
lengths. We write this as AB−−−AB_ ≅ EF−−−EF_ or AB = EF.
If two angles have the same measure, they are congruent. Also, if two angles are congruent, their
measures are the same. If two angles are congruent, then the lengths of their arms do not matter.
e.g. Two angles, say, ∠PQR and ∠XYZ, are congruent if their measures are equal. We write this
as ∠PQR ≅ ∠XYZ or as ∠PQR = ∠XYZ. However, commonly, we write ∠PQR ≅ ∠XYZ.
Two circles of equal radii are congruent. Two squares of equal sides are congruent.
Criteria for Congruence of Triangles
Congruence of triangles
Consider triangles ABC and XYZ. Cut triangle ABC and place it over XYZ. The two triangles
cover each other exactly, and they are of the same shape and size. Also notice that A falls on X,
B on Y, and C on Z. Also, side AB falls along XY, side BC along YZ, and side AC along XZ.
So, we can say that triangle ABC is congruent to triangle XYZ. Symbolically, it is represented as
ΔABC ≅ ΔXYZ.
So, in general, we can say that two triangles are congruent if all the sides and all the angles of
one triangle are equal to the corresponding sides and angles of the other triangle.
Order of the letters in the names of the congruent triangles shows the corresponding relationship.
In two congruent triangles ABC and XYZ, the corresponding vertices are A and X, B and Y, and
C and Z, that is, A corresponds to X, B to Y, and C to Z. Similarly, the corresponding sides are
AB and XY, BC and YZ, and AC and XZ. Also, angle A corresponds to X, B to Y, and C to Z.
So, we write ABC corresponds to XYZ.
Criteria for congruence of triangles
Two triangles can be congruent if three of their corresponding parts are equal.
SSS congruence criterion
Two triangles are congruent if three sides of one triangle are equal to the three corresponding
sides of the other triangle.
SAS congruence criterion
Two triangles are congruent if two sides and the included angle of one triangle are equal to the
corresponding two sides and the included angle of the other triangle.
RHS congruence criterion
Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to
the hypotenuse and the corresponding side of the other triangle.
ASA congruence criterion
Two triangles are congruent if two angles and the included side of one triangle are equal to the
corresponding two angles and the included side of the other triangle.
Two triangles with equal corresponding angles may not be congruent. So, there is no such thing
as AAA congruence of triangles. Two congruent triangles have equal areas and equal perimeters.
Ratios and Proportions
Ratio
Ratios are used to compare quantities. To compare two quantities, the units of the quantities must
be the same. Ratios help us to compare quantities and determine the relation between them. We
write ratios in the form of fractions and then compare them by converting them to like fractions.
If these like fractions are equal, then the ratios are said to be equivalent.
e.g. Cost of 6 pens is Rs 90. What would be the cost of 10 such pens?
Solution: Cost of 6 pens = Rs 90
Cost of 1 pen = 90 ÷ 6 = Rs 15
Hence, cost of 10 pens = 10 × 15 = Rs 150.
Proportion
When two ratios are equivalent, the four quantities are said to be in proportion.
Ratio and proportion problems can be solved by using two methods, the unitary method and
equating the ratios to make proportions, and then solving the equation.
Unitary method
Unitary method is the method of finding the value of one unit (unit rate) at first and then the
value of required number of units.
Percentages
Percentage is another method used to compare quantities. Percent is derived from the Latin word
‘per centum’, which means per hundred. Percentage is the numerator, of a fraction, whose
denominator is hundred. Percent is represented by the symbol - %.
e.g. 2110021100 or 21%
Percentages
Percentage is the numerator of a fraction, whose denominator is hundred. If the denominator in
the fraction is not hundred we multiply the fraction with hundred percent to get percentage.
Percentages are also used for comparing quantities.
Converting fractional numbers into percentage
To convert a fraction into a percentage, multiply it by hundred and then place the % symbol.
e.g. 4545 = 4545 × 100 = 80%
Percentages related to proper fractions are less than 100, whereas percentages related to improper
fractions are more than 100.
Converting decimals into percentage
To convert a decimal into percentage, multiply the decimal by hundred and then place the %
symbol.
e.g. 0.25
= 2510025100 × 100
= 25%
Conversions
A given percentage can be converted into fractions and decimals. Also, a decimal can be
converted as percentage.
To convert a percentage into a fraction
Remove the percent sign, and then divide the number by hundred.
e.g. 20% = 2010020100 = 1515.
To convert a percentage into a decimal
Step 1: Remove the percent sign.
Step 2: Divide the number by 100, or move the decimal point two places to the left in the
numerator.
e.g. 4% = 41004100 = 0.04
To convert a decimal into a percentage
Step1: Convert the decimal into a fraction.
Step 2: Multiply the fraction by hundred.
Step 3: Put a percent sign next to the number.
Else, shift the decimal point two places to the right.
e.g. 0.25
= 2510025100
= 2510025100 × 100
= 25 %
Application of Percentages
Percentages are helpful in comparison. The increase or decrease in a certain quantity can be
expressed as percentage increase or decrease.
Percentage increase/decrease = Amount of change (Increase or Decrease)Original amount or
baseAmount of change (Increase or Decrease)Original amount or base ×100
Profit and loss
Profit is the money gained on selling an item. Loss is the money lost on selling an item. If the
selling price of an item is more than the cost price, then a profit is incurred. If the cost price of an
item is more than the selling price, then a loss is incurred. When the cost price is equal to the
selling price, it is called a no-profit no-loss situation.
Profit = Selling price – Cost price
Loss = Cost price – Selling price
CP = SP ⇒ No Profit no Loss
Profit percentage or loss percentage is always calculated on the cost price.
Profit percentage = ProfitCost priceProfitCost price × 100
Loss percentage = LossCost priceLossCost price × 100
Simple interest
Interest is the fee paid as the cost of borrowing on the borrowed money. The way of calculating
interest, where the principal is not changed, is known as "simple interest". As the number of
years increases, interest will also increase.
If P denotes the principal, R denotes the rate of interest and T denotes the time period, then the
simple interest I paid for T years is P × T × R100P × T × R100.
Sum that is returned at the end of the time period is called the amount (A). Amount can be
obtained by adding the sum borrowed (principal) and the interest.
Amount = Principal + Interest.
Introduction to Rational Numbers
All numbers, including whole numbers, integers, fractions and decimal numbers, can be written
in the NumeratorDenominator NumeratorDenominator form.
Rational number
A rational number is a number that can be written in the form p/q, where p and q are integers and
q ≠ 0. The denominator of a rational number can never be zero.
e.g. 911911, 5858, 712712
A rational number is positive if its numerator and denominator are both either positive integers
or negative integers.
e.g. 2525, 3434, -7-10-7-10, -5-11-5-11
If either the numerator or the denominator of a rational number is a negative integer, then it is a
negative rational number.
e.g. -25-25, 3-43-4, 7-107-10, 5-115-11
Representation of rational numbers on the number line
The rational number zero is neither negative nor positive. Positive rational numbers are
represented to the right of zero on the number line. Negative rational numbers are represented to
the left of zero on the number line.
A rational number obtained by multiplying or dividing both the numerator and the denominator
of a rational number by the same non-zero integer, is said to be the equivalent form of the given
rational number.
Rational numbers in Standard form
A rational number is said to be in its standard form if its numerator and denominator have no
common factor other than 1, and its denominator is a positive integer.
To reduce a rational number to its standard form, divide its numerator and denominator by their
highest common factor (HCF). To find the standard form of a rational number with a negative
integer as the denominator, divide its numerator and denominator by their HCF with a minus
sign.
Comparison of Rational Numbers
Among the positive rational numbers with the same denominator, the number with the greatest
numerator is the largest. It is easy to compare the rational numbers with same denominators.
e.g. 28302830 > 26302630 > 21302130.
A negative rational number is to the left of zero whereas a positive rational number is to the right
of zero on a number line. So, a positive rational number is always greater than a negative rational
number.
To compare two negative rational numbers with the same denominator, their numerators are
compared ignoring the minus sign. The number with the greatest numerator is the smallest.
e.g. –710710 < – 310310; – 6767 < – 4747
To compare rational numbers with different denominators, they are converted into equivalent
rational numbers with the same denominator, which is equal to the LCM of their denominators.
There are unlimited number of rational numbers between two rational numbers. To find a
rational number between the given rational numbers, they are converted to rational numbers with
same denominators.
Operations on Rational Numbers
Addition of Rational numbers
The sum of two rational numbers with the same denominator is a rational number whose
numerator is the sum of the numerators of the rational numbers with the same denominator.
To add rational numbers with different denominators, they are converted into equivalent rational
numbers with the same denominator.
Additive inverse of a Rational number
Two rational numbers whose sum is zero are called the additive inverses of each other.
e.g. -720-720 is the additive inverse of 720720 and 720720 is the additive inverse of -720-720.
Subtraction of Rational numbers
The difference between two rational numbers with the same denominator is a rational number
whose numerator is the difference of the numerators of the rational numbers with the same
denominator.
To subtract rational numbers with different denominators, they are converted into equivalent
rational numbers with the same denominator.
Multiplication of Rational numbers
The numerator and denominator of the product of two rational numbers are equal to the product
of their individual numerators and denominators.
The numerator of the product of a rational number and an integer is equal to the product of the
numerator and the integer with the same denominator.
Reciprocal of a Rational number
Two rational numbers whose product is 1 are called reciprocals of each other. A rational number
and its reciprocal will always have the same sign.
e.g. -136-136 × 6-136-13 = 1
Division of Rational numbers
To divide one rational number by another, first number is multiplied with the reciprocal of the
second number.
e.g. 518518 ÷ 2929 = 518518 × 9292 = 54
Construction of Triangles
Any one of the following sets of measurements are required to construct a triangle• Length of the three sides
• Two sides and the included angle
• Two angles and the included side
• Length of the hypotenuse and one side in case of a right-angled triangle.
Construction of a triangle when measurements of its three sides are given
Construct ΔABC, when AB = 6 cm, BC = 7 cm and CA = 9 cm.
Steps of construction:
Step 1: Draw line segment BC = 7 cm.
Step 2: Draw an arc with B as the centre and the radius equal to 6 cm.
Step 3: Draw an arc with C as the centre and the radius equal to 9 cm.
Step 4: Name the point of intersection of these two arcs as A.
Step 5: Join points A and B, and points A and C.
Triangle ABC is the required triangle.
Construction of a triangle when measurements of two sides and the included angle are
given
Construct ΔPQR, when PQ = 4 cm, QR = 6 cm and ∠PQR = 60°.
Steps of construction:
Step 1: Draw line segment QR = 6 cm.
Step 2: Construct an angle of 60° at point Q.
Step 3: Draw an arc on the ray QX with Q as the centre and the radius equal to 4 cm.
Step 4: Name the point where the arc cuts ray QX, as P.
Step 5: Join points P and R.
Triangle PQR is the required triangle.
Construction of a triangle, when two angles and the included side are given
Construct ΔXYZ, when ∠ZXY = 40°, ∠XYZ = 95° and the included side XY = 8 cm.
Steps of construction:
Step 1: Draw line segment XY = 8 cm.
Step 2: Construct an angle of 40° at X with XY.
Step 3: Construct another angle of 95° at Y with YX.
Step 4: Name the point of intersection of the two rays as Z.
Triangle XYZ is the required triangle.
Construction of a right-angled triangle, when the length of one side and the hypotenuse are
given
Construct a right-angled triangle LMN, with hypotenuse LN = 8 cm and side MN = 5 cm.
Steps of construction:
Step 1: Draw line 'l'.
Step 2: Mark a point on 'l' and name it M.
Step 3: Draw a line segment MN = 5 cm on 'l' .
Step 4: Construct a right angle XMN at M.
Step 5: Draw an arc with N as the centre and radius equal to 8 cm, such that it intersects MX.
Step 6: Mark the point of intersection as L.
Step 7: Join points L and N.
Triangle LMN is the required triangle.
Construction of Parallel Lines
Two lines in a plane that never meet each other at any point are said to be parallel to each other.
Any line intersecting a pair of parallel lines is called a transversal.
Properties of angles formed by parallel lines and transversal:
• All pairs of alternate interior angles are equal.
• All pairs of corresponding angles are equal.
• All pairs of alternate exterior angles are equal.
• The interior angles formed on the same side of the transversal are supplementary (the sum
of their measures is 180°).
Construction of a parallel line using the alternate interior angle property
Step 1: Draw line 'l' and point A outside it.
Step 2: Mark point B on line 'l'.
Step 3: Draw line 'n' joining point A and point B.
Step 4: Draw an arc with B as the centre, such that it intersects line 'l' at D and line 'n' at E.
Step 5: Draw another arc with the same radius and A as the centre, such that it intersects line 'n'
at F. Ensure that arc drawn from A cuts the line 'n' between A and B.
Step 6: Draw another arc with F as the centre and distance DE as the radius.
Step 7: Mark the point of intersection of this arc and the previous arc as G.
Step 8: Draw line 'm' passing through points A and G.
Line 'm' is the required parallel line.
Verification of the construction
If the pair of alternate interior angles are equal in measure, then line 'm' is parallel to line 'l'.
Construction of a parallel line using the corresponding angle property
Step 1: Draw line 'l' and point P outside it.
Step 2: Mark point Q on line 'l'.
Step 3: Draw line 'n' joining point P and point Q.
Step 4: Draw an arc with Q as the centre, such that it intersects line 'l' at R and line 'n' at S.
Step 5: Draw another arc with the same radius and P as the centre, such that it intersects line 'n' at
X. Ensure that arc drawn from P cuts the line 'n' outside QP.
Step 6: Draw another arc with X as the centre and distance RS as the radius, such that it
intersects the previous at Y.
Step 7: Draw line 'm' passing through points P and Y.
Line 'm' is the required parallel line.
Verification of the construction
If the pair of corresponding angles are equal in measure, then line 'm' is parallel to line 'l'.
Plane Figures
Perimeter of a closed figure is the distance around it, whereas area is the region enclosed by
closed figure.
Perimeter of a regular polygon = number of sides x length of one side.
Area and perimeter of a rectangle
The perimeter of a rectangle is twice the sum of the lengths of its adjacent sides.
Perimeter of a rectangle of length 'l' units and breadth 'b' units = 2(l + b).
The area of a rectangle is the product of its length and breadth.
Area of a rectangle of length 'l' units and breadth 'b' units = l × b.
The perimeter of rectangle ABCD = 2(AB + BC).
Area of rectangle ABCD = AB x BC.
Each diagonal of a rectangle divides it into two triangles that are equal in area.
Area and perimeter of a square
The perimeter of a square with side s units is the four times the length of its side.
Perimeter of a square with side s units = 4 × s
The area of a square with side s is is equal to side multiplied by side.
Area of a square with side s units = s × s
The perimeter of square ABCD = 4AB or 4BC or 4CD or 4DA.
Area of square ABCD = AB2 or BC2 or CD2 or DA2.
The diagonals of a square divide it into four triangles that are equal in area. A rectangle and a
square having the same perimeter need not have the same area. If the perimeter of a figure
increases it is not necessary that its area also increases.
Area and perimeter of a triangle
The perimeter of a triangle is the sum of the lengths of its sides.
Perimeter of a triangle with sides a, b and c = (a + b + c).
The area of a triangle is the space enclosed by its three sides.
Area of a triangle is half of the product of its base and the corresponding altitude.
Area of a triangle with b as the base and h as the altitude = 1212 × bh.
Triangles equal in area need not be congruent, but all congruent triangles are equal in area.
Area and perimeter of a parallelogram
The perimeter of a parallelogram is twice the sum of the lengths of the adjacent sides.
The area of a parallelogram is the product of its base and the corresponding altitude.
Area of a parallelogram with b as the base and h as the altitude = b × h.
Any side of a parallelogram can be considered as the base. The perpendicular drawn on that side
from the opposite vertex is known as the height (altitude).
The perimeter of parallelogram ABCD = 2(AB + BC)
Area of parallelogram ABCD = (AB x DE) or (AD x BF).
A parallelogram in which the adjacent sides are equal is called a rhombus.
The perimeter and area of a rhombus can be calculated using the same formulae as that for a
parallelogram.
Conversion of units
1cm = 10 mm
1 cm2 = 100 mm2
1 m2 = 10000 cm2
1 hectare = 10,000 m2
Circles
A circle is defined as a collection of points on a plane that are at an equal distance from a fixed
point on the plane. The fixed point is called the centre of the circle.
Any line segment that passes through the centre of a circle and whose end points are on the circle
is called its diameter.
Any line segment from the centre of the circle to its circumference is called the radius of the
circle.
The diameter of a circle is two times the radius.
Circumference of a circle
The distance around a circular region is known as its circumference.
Ratio of circumference and diameter of a circle is denoted by the Greek symbol π.
π is an irrational number, whose value is approximately equal to 227227 or 3.14
Circumference of a circle = 2πr, where r is the radius of the circle or
Circumference of a circle = πd, where d is the diameter of the circle.
Circumference = Diameter x 3.14
Area of a circle
The area of a circle is the region enclosed in the circle.
The area of a circle can be calculated by using the formula is πr2, if radius r is given; πd24πd24,
if diameter d is given; C24πC24π, if circumference C is given.
Concentric circles
Circles with the same centre but different radii are called concentric circles.
Area between two concentric circles = Area of outer circle – Area of inner circle.
Conversion of units
To convert from a unit of area to its smaller unit of area, we multiply. To convert from a unit of
area to a larger unit of area, we divide.
1 cm = 10 mm
1 m = 100 cm
1 km = 1000 m
1 cm2 = 100 mm2
1 m2 = 10000 cm2
1 km2 = 1000000 m2
1 hectare = 10000 m2
Understanding Algebraic Expressions
Expressions that contain only constants are called numeric or arithmetic expressions. It is a set of
numerical values that are separated by the four mathematical operations, addition, subtraction,
multiplication and division.
e.g. 9 + 8, 5 – 3
Expressions that contain constants and variables, or just variables, are called algebraic
expressions. Variables and constants are combined using mathematical operations to form an
algebraic expression.
e.g. x – 5, 3b – 6
While writing algebraic expressions, we do not write the sign of multiplication. An algebraic
expression containing only variables also has the constant 1 associated with it. The parts of an
algebraic expression added to form the expression are called its terms.
e.g. The number of terms in the expression x2 + 3x + 5 is 3.
A term that contains variables is called a variable term. A term that contains only a number is
called a constant term.
The constants and the variables whose product makes a term of an algebraic expression, are
called the factors of the term. The factors of a constant term in an algebraic expression are not
considered. The numerical factor of a variable term is called its coefficient. If the coefficient of a
term is 1, it is usually omitted. If the coefficient of a term is –1, it is indicated by only the minus
sign. The variable factors of a term are called its algebraic factors.
Terms that have different algebraic factors are called unlike terms. Terms that have the same
algebraic factors are called like terms. We can compare only the like terms.The unlike terms can
not be compared.
Algebraic expressions that contain only one term are called monomials. Algebraic expressions
that contain two unlike terms are called binomials. Algebraic expressions that contain three
unlike terms are called trinomials. All algebraic expressions that have one or more terms are
called polynomials. Therefore, monomials, binomials and trinomials are also polynomials.
Operations on Algebraic Expressions
Addition of algebraic expressions
To add algebraic expressions, rearrange the terms in the sum of the given algebraic expressions,
so that their like terms and constants are grouped together. While rearranging terms, move them
with the correct plus (+) or minus (–) sign before them.
To add like terms in an algebraic expression, multiply the sum of their coefficients with their
common algebraic factors.
e.g. Add 5x2y + 6 and 2x2y – 11.
Sol: (5x2y + 6) + (2x2y – 11)
= 5x2y + 6 + 2x2y – 11
= 5x2y + 2x2y + 6 – 11
= (5 + 2)x2y + 6 – 11
= 7x2y – 5.
Subtraction of algebraic expressions

To subtract algebraic expressions
Change the signs of the terms of the expression being subtracted.




Rearrange the terms in the difference of the given algebraic expressions, so that their like terms
and constants are grouped together.
While rearranging terms, move them with the correct signs before them.
Multiply the difference of their coefficients with their common algebraic factors.
Unlike terms remain unchanged in the sum or difference of algebraic expressions.
e.g. Subtract 2xy – 3x 2y – 4 from 2x2y – 3xy + 4y + 5.
= (2x2y – 3xy + 4y + 5) – (2xy – 3x2y – 4)
= 2x2y – 3xy + 4y + 5 – 2xy + 3x2y + 4
= 2x2 y + 3x2y – 3xy – 2xy + 4y + 5 + 4
= (2 + 3)x2y – 3xy – 2xy + 4y + 5 + 4
= 5x2 y – 5xy + 4y + 9.
Application of Algebraic Expressions
Algebraic expressions can be used to represent number patterns.
Ex: Table showing the relation between the number of cones and the number of ice-cream
scoops.
Number of
cones(n)
Number of ice-cream
scoops (2n)
1
2
2
4
3
6
8
16
15
30
Thus, we can find the value of an algebraic expression if the values of all the variables in the
expression are known.
e.g. Find the value of the expression 3x2y – 2xy2 + 2xy for x = 2 and y = –2.
Sol:
3x2y – 2xy2 + 2xy .
Putting x = 2 and y = –2 in the given expression,
3x2y – 2xy2 + 2xy
= 3×(2)2×(–2) – 2×(2)×(–2)2 + 2×(2)×(–2)
= 3×4×(–2) – 4×4 + 4×(–2)
= – 24 –16 – 8
= – 48.
Formulas and rules such as the perimeter and area for different geometrical figures are written in
a concise and general form using simple, and easy-to-remember algebraic expressions.
If 's' represents the side of a square, then its perimeter is '4s' and area is 's2'.
If 'l' represents the length and 'b' represents the breadth of a rectangle, then its perimeter is '2(l +
b)' and area is 'l × b'.
Area of a triangle with base 'b' and the corresponding altitude 'h' is '1212 × base × height'.
Perimeter of an equilateral triangle with the length of the side as 'a' units is '3a'.
Exponents and Powers
An exponent or power is a mathematical representation that indicates the number of times that a
number is multiplied by itself.
If a number is multiplied by itself m times, then it can be written as: a x a x a x a x a...m times =
am. Here, a is called the base, and m is called the exponent or power or index.
Numbers raised to the power of two are called square numbers. Square numbers are also read as
two-square, three-square, four-square, five-square, and so on. Numbers raised to the power of
three are called cube numbers. Cube numbers are also read as two-cube, three-cube, four-cube,
five-cube, and so on.
Negative numbers can also be written using exponents. If an = b, where a b and n are integers,
then an is called the exponential form of b. When numbers are expressed as the product of the
powers of their prime factors, we get the prime factor product form. The order of factors in the
prime factor product form of a number can be interchanged without changing the value of the
number.
e.g. ax × by = by × ax.
When numbers are expressed using powers of 10, we get the expanded form of numbers. The
value of an exponential number with a negative base raised to the power of an even number is
positive. The value of an exponential number with a negative base raised to the power of an odd
number is negative.
(–1)odd number = –1
(–1)even number = + 1
If the base of two exponential numbers is the same, then the number with the greater exponent is
greater than the number with the smaller exponent.
A number can be expressed as a decimal number between 1.0 and 10.0, including 1.0, multiplied
by a power of 10. Such a form of a number is known as its standard form. Large numbers when
written in the standard form is much easier to read, understand and compare.
Laws of Exponents
Multiplication of Powers with the Same Base
When numbers with the same base are multiplied, the power of the product is equal to the sum of
the powers of the numbers.
If 'a' is a non-zero integer, and 'm' and 'n' are whole numbers then, am × an = am+n.
Division of Powers with the Same Base
When numbers with the same base are divided, then the power of the quotient is equal to the
difference between the powers of the dividend and the divisor.
If 'a' is a non-zero integer, and 'm' and 'n' are whole numbers then, am ÷ an = am-n.
Power of a Power
If 'a' is any non-zero integer, and ‘m’and ‘n’ are whole numbers then, (am)n = amn.
Multiplication of Powers with the Same Exponent
If 'a' is any non-zero integer, and ‘m’ is a whole number then, am × bm = (ab)m.
Division of Powers with the Same Exponent
If a and b are any non-zero integers and m is a whole number then, am ÷ bm = (abab)m.
Numbers with an Exponent of Zero
For any non-zero integer a, a0 = 1.
Introduction to Solid Shapes
All two-dimensional figures have only length and breadth.
e.g. A paper has only length and breadth, and hence, it is classified as a plane or two-dimensional
figure.
Three-dimensional solid shapes have length, breadth and height.
e.g. a box is in three-dimensional shape.
Faces
The flat surface of a three dimensional shape is called a Face.
Edges
Two faces of a solid shape meet to form an Edge.
Vertices
The points where three edges meet are called vertices.
The table shows the number of the faces, edges and vertices of some shapes.
Solid shape
Face (F)
Edge (E)
Vertices (V)
3
2
0
2
1
1
4
6
4
Nets of three-dimensional shapes
The net of a three-dimensional solid is a two-dimensional skeleton outline, which, when folded,
results in the three-dimensional shape.
Drawing solid shapes on a flat surface
Solid shapes can be drawn on a flat surface, which is known as the two-dimensional
representation of a three-dimensional solid.
Sketches of solids are of two types: oblique and isometric.
Oblique sketches are drawn on squared paper. The oblique sketch gives the visual representation
of a solid shaped object but does not the represent the actual dimensions.
Isometric sketches are drawn on dotted or isometric sheets and have the exact measurements of
solids.
The two-dimensional surface on which we draw an image is usually flat. So, when we try
drawing a solid shape on a paper or a board, the image appears a little distorted. However, this is
just an optical illusion. When look at the picture of the cube. Not all the lines forming the cube
are of equal length. Also, we are unable to see all the faces of the cube. In spite of this illusion,
we can make out that the image is of a cube.
Such skeletons of solids are called oblique sketches. They can be drawn using a squared paper.
The first step is to draw the front face of the cube. Then the opposite face of the cube are drawn.
This face should also be of the same size as that of the first square. The number of squares is
used as a reference for maintaining the size. Now, the corresponding corners of the squares are
joined. And finally, the edges that cannot be seen are drawn, with dotted lines. We can see that
the front face and the opposite face of the cube are of the same size. Also, the edges appear
equal, though we do not draw them of the same length.
All of us must have solved puzzles on isometric sheets at some point of time. An isometric sheet
divides a screen into small equilateral triangles made of dots. Using this sheet, we can draw
sketches with measurements that agree with that of a given solid. Let’s try to draw the sketch of
a cuboid of length three units, breadth two units and height two units.
First, a rectangle of length three units and breadth two units is drawn. Then four parallel line
segments, each of length two units are drawn starting from the four corners of the rectangle.
Finally, the matching corners with appropriate line segments are connected. The isometric sketch
of the cuboid is formed. Observe that the measurements are of the exact size, which is not the
case with oblique sketches.
Viewing the Different Section of Solids
Visualising solid shapes is a very useful skill. We can see the hidden parts of a solid shape. For
example, when a cuboid with a square face is cut vertically, then each face is a square. The face
is a cross section of the cuboid.
Three-dimensional objects or solids have length, breadth and height, and look different from
different points of view.
Sections of solid shapes can be viewed using three ways:
• Viewing the cross sections
• Using shadows
• Viewing at certain angles
A solid can be viewed from different angles. Viewing a solid from the front, side and top are the
three most common ways of viewing it.
Cutting or slicing a solid will result in its cross-section, which is also one way of viewing it.
Observing the two-dimensional shadow of a three-dimensional solid is another method of
viewing a solid.
Shadows of solids are of different sizes depending on the position of the solid and the position of
the source of light.