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IX_Maths Chapter Notes
1
IX
Mathematics
Chapter 1: Number Systems
Chapter Notes
Key Concepts
1.
Numbers 1, 2, 3……., which are used for counting are called Natural
numbers and are denoted by N.
2.
0 when included with the natural numbers form a new set of numbers
called Whole number denoted by W
3.
-1,-2,-3……………..- are the negative of natural numbers.
4.
The negative of natural numbers, 0 and the natural number together
constitutes integers denoted by Z.
5.
The numbers which can be represented in the form of p/q where
q  0 and p and q are integers are called Rational numbers. Rational
numbers are denoted by Q. If p and q are coprime then the rational
number is in its simplest form.
6.
Irrational numbers are the numbers which are non-terminating and
non-repeating.
7.
Rational and irrational numbers together constitute Real numbers
and it is denoted by R.
8.
Equivalent rational numbers (or fractions) have same (equal)
values when written in the simplest form.
9.
Terminating fractions are the fractions which leaves remainder 0 on
division.
10.
Recurring fractions are the fractions which never leave a remainder
0 on division.
11.
There are infinitely many rational numbers between any two rational
numbers.
12.
If Prime factors of the denominator are 2 or 5 or both only. Then the
number is terminating else repeating/recurring.
2
13.
Two numbers p & q are said to be co-prime if, numbers p & q have no
common factors other than 1.
14.
The decimal expansion of rational number is either terminating or
non-terminating recurring
15.
The decimal expansion of an irrational number is non-terminating,
non-recurring.
16.
Real numbers satisfy the commutative, associate and distributive
law of addition and multiplication.
17.
Commutative law of addition: If a and b are two real numbers then,
a+b=b+a
19. Commutative law of multiplication: If a and b are two real numbers
then, a. b = b. a
20.
Associative law of addition: If a, b and c are real numbers then,
a + (b + c) = (a + b) + c
21.
Associative law of multiplication: If a, b and c are real numbers
then, a. (b. c) = (a. b). c
22.
Distributive of multiplication with respect to addition: If a, b and
c are real numbers then, a. (b+ c) = a. b + a. c
23.
Removing the radical sign from the denominator is called
rationalisation of denominator.
24.
The multiplication factor used for rationalising the denominator is called
the rationalising factor.
25.
The exponent is the number of times the base is multiplied by itself.
26.
In the exponential representation am , a is called the base and m
is called the exponent or power.
3
27.
If a number is to the left of the number on the number line, it is less
than the other number. If it is to the right, then it is greater than the
number.
28.
There is one to one correspondence between the set of real
numbers and the set of point on the number line.
30.
Irrational numbers like
2,
3,
5…
n , for any positive integer
n can be represented on number line by using Pythagoras theorem.
31.
The process of visualisation of representation of numbers on the
number line through a magnifying glass is known as the process of
successive magnification.
Key Formulae:
1. Rational number between two numbers x and y =
2.
xy
2
Irrational number between two numbers x and y
 xy, if x andybothareirrationalnumbers

 xy, if xisrationalnumber andyisirrationalnumber 


 xy, if x  yisnot aperfect square andx,ybotharerationalnumbers
3.
Irrational number between two rational number x and y =


xy , if and
only if x  y is not a perfect square.
4.
Irrational number between a rational number x and irrational number
y=
xy
5.
2
xy
 x = y2
6.
3
xy
 x = y3
7.
n
xy
 x = yn
4
8.
Square root identities: For real numbers a>0 and b>0
ab 

a






a

b

b
(
a 
(
a 
(a 

a b
(
b)(
b)(
c 
b)(a 
a 
a 
b)  a  b
d) 
ac 
b)  a2  b
b)2  a  b  2

bc 
ad 
bd


ab

1
n
n
a  a  an
where a >0 and is a real number and n is positive integer.
9.
m
m
10.
a
n

n a   n am
where a, n > 0 and ‘a’ is a real number, m and n co prime integers.
11.
a0  1 , where a is a real number.
12.
Pythagoras Theorem: (AB) 2 + (BC) 2 = (AC) 2
where AC is hypotenuse, AB and BC are the sides of the right triangle.
13.
x
 x  1 2


 x  1 2


2 
2 


1
IX Math
Ch 2: Polynomials
Chapter Notes
Top Definitions
1.
A polynomial p(x) in one variable x is an algebraic expression in x of
the form
p(x) = a xn  a xn1  a xn2  ........  a x 2  a x  a , where
n
n1
n2
2
1
0
(i) a0 ,a1 ,a2 ......an are constants
(ii) x 0 ,x1 ,x 2 ......xn are variables
(iii) a0 ,a1 ,a2 ......an are respectively the coefficients of x
n
n1
(iv) Each of an x  an 1x ,an
called a term of a polynomial.
n2
2x
,........a2 x
2
0
,x1 ,x
2
......xn .
,a1x,a0 , with an  0, is
2.
A leading term is the term of highest degree.
3.
Degree of a polynomial is the degree of the leading term.
4.
A polynomial with one term is called a monomial.
5.
A polynomial with two terms is called a binomial.
6.
A polynomial with three terms is called a trinomial.
7.
A polynomial of degree 1 is called a linear polynomial. It is of the form
ax+b. For example: x-2, 4y+89, 3x-z.
8.
A polynomial of degree 2 is called a quadratic polynomial. It is of the
form ax2 + bx + c. where a, b, c are real numbers and a 0 For
example: x 2  2x  5 etc.
9.
A polynomial of degree 3 is called a cubic polynomial and has the
general form ax3 + bx2 + cx +d. For example: x 3  2x 2  2x  5 etc.
10.
A bi-quadratic polynomial p(x) is a polynomial of degree 4 which can
be reduced to quadratic polynomial in the variable z = x2 by
substitution.
2
11.
The zero polynomial is a polynomial in which the coefficients of all the
terms of the variable are zero.
12.
Remainder theorem: Let p(x) be any polynomial of degree greater
than or equal to one and let a be any real number. If p(x) is divided by
the linear polynomial x – a, then remainder is p(a).
13.
Factor Theorem: If p(x) is a polynomial of degree n≥ 1and a is any
real number then (x-a) is a factor of p(x), if p(a) =0.
14.
Converse of Factor Theorem: If p(x) is a polynomial of degree n≥ 1and
a is any real number then p(a) =0 if (x-a) is a factor of p(x).
15.
An algebraic identity is an algebraic equation which is true for all
values of the variables occurring in it.
Top Concepts
1.
The degree of non-zero constant polynomial is zero.
2.
A real number ‘a’ is a zero/ root of a polynomial p(x) if p (a) = 0.
3.
The number of real zeroes of a polynomial is less than or equal to the
degree of polynomial.
4.
Degree of zero polynomial is not defined.
5.
A non zero constant polynomial has no zero.
6.
Every real number is a zero of a zero polynomial.
7.
Division algorithm: If p(x) and g(x) are the 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 (x) = g(x) q(x) + r(x)
where, r(x) =0 or degree of r(x) < degree of g(x).
8.
If the polynomial p(x) is divided by (x+a), the remainder is given by
the value of p (-a).
9.
If the polynomial p(x) is divided by (x-a), the remainder is given by
the value of p (a).
3
10.
If p (x) is divided by ax + b = 0; a  0, the remainder is given by
p  b  ; a  0.
 a

11.

If p (x) is divided by ax - b = 0 , a  0 , the remainder is given by
p  b  ; a  0.
a


12.
A quadratic polynomial ax2 + bx+ c is factorised by splitting the middle
term bx as px +qx so that pq =ac.
13.
The quadratic polynomial ax2 + bx+ c will have real roots if and only
if b2-4ac ≥ 0.
14.
For applying factor theorem the divisor should be either a linear
polynomial of the form x-a or it should be reducible to a linear
polynomial.
Top Formulae
1.
Quadratic identities:
a.
b.
c.
d.
x  y 2  x2  2xy  y2
x  y 2  x2  2xy  y2
x  y (x  y)  x 2  y2
x  a  (x  b)  x 2  (a  b)x  ab
x  y  z 2  x2  y2  z2 2xy  2yz  2zx
e.
Here x, y, z are variables and a, b are constants
2.
Cubic identities:
a.
b.
x  y 3
x  y 3
 x3  y3  3xy(x  y)
 x3  y3  3xy(x  y)
c. x3  y3  (x  y)(x2  xy  y2 )
d. x3  y3  (x  y)(x2  xy  y2 )
e. x3  y3  z3  3xyz  (x  y  z)(x2  y2  z2  xy  yz  zx)
f. If x  y  z  0 then x3  y3  z3  3xyz
Here, x, y & z are variables.
1
IX
Mathematics
Chapter 3: Coordinate Geometry
Points to Remember
Key Concepts
1.
Two perpendicular number lines intersecting at point zero are called
coordinate axes. The horizontal number line is the x-axis (denoted
by X’OX) and the vertical one is the y-axis (denoted by Y’OY).
2.
The point of intersection of x axis and y axis is called origin and
denoted by ‘O’.
3.
Cartesian plane is a plane obtained by putting the coordinate axes
perpendicular to each other in the plane. It is also called coordinate
plane or xy plane.
4.
The x-coordinate of a point is its perpendicular distance from y axis.
5.
The y-coordinate of a point is its perpendicular distance from x axis.
6.
The point where the x axis and the y axis intersect is represented by
coordinate points (0, 0) and is called the origin. It is denoted by ‘O’
on a Cartesian plane.
7.
The abscissa of a point is the x-coordinate of the point.
8.
The ordinate of a point is the y-coordinate of the point.
9.
If the abscissa of a point is x and the ordinate of the point is y, then
(x, y) are called the coordinates of the point.
10.
The axes divide the Cartesian plane into four parts called the
quadrants (one fourth part), numbered I, II, III and IV anticlockwise
from OX.
11.
The origin O has zero distance from both the axes.
12.
The coordinate of a point on the x axis are of the form (x,0) and that
of the point on y axis are (0,y)
13.
Sign of coordinates depicts the quadrant in which it lies. The
coordinates of a point are of the form (+, +) in the first quadrant,
(-, +) in the second quadrant, (-,-) in the third quadrant and (+,-) in
the fourth quadrant.
2
14.
To plot a point P (3, 4) in the Cartesian plane. Start from origin count
3 units on the positive x axis then move 4 units towards positive y axis
and mark the point P.
15.
If x ≠ y, then (x,y)≠(y,x) and if (x,y) = (y,x), then x=y.
1
IX
Mathematics
Chapter 4: Linear Equations in Two Variables
Chapter Notes
Top Definitions
1.
An equation of the form ax + by + c = 0, where a, b and c are real
numbers, such that a and b are not both zero, is called a linear equation
in two variables.
2.
A linear equation in two variables is represented geometrically by a
straight line the points of which make up the collection of solutions of
equation. This is called the graph of the linear equation.
Top Concepts
1.
A linear equation in two variables has infinitely many solutions.
2.
The graph of every linear equation in two variables is a straight line.
3.
x = 0 is the equation of the y – axis and y = 0 is the equation of the
x–axis.
4.
The graph of x = k is a straight line parallel to the y –axis.
5.
The graph of y = k is a straight line parallel to the x – axis.
6.
An equation of the type y = mx represents a line passing through the
origin, where m is a real number.
7.
Every point on the line satisfies the equation of the line and every
solution of the equation is a point on the line.
8.
The solution of a linear equation is not effected when:
(i) The same number is added or subtracted from both the side of
an equation.
(ii) Multiplying or dividing both the sides of the equation by the
same non zero number.
2
Top Diagrams
1.
Graph of a line passing through the origin.
2.
Graph of a line parallel to x axis.
3
3.
Graph of a line parallel to y axis.
1
IX
Mathematics
Chapter 4: Linear Equations in Two Variables
Chapter Notes
Top Definitions
1.
An equation of the form ax + by + c = 0, where a, b and c are real
numbers, such that a and b are not both zero, is called a linear equation
in two variables.
2.
A linear equation in two variables is represented geometrically by a
straight line the points of which make up the collection of solutions of
equation. This is called the graph of the linear equation.
Top Concepts
1.
A linear equation in two variables has infinitely many solutions.
2.
The graph of every linear equation in two variables is a straight line.
3.
x = 0 is the equation of the y – axis and y = 0 is the equation of the
x–axis.
4.
The graph of x = k is a straight line parallel to the y –axis.
5.
The graph of y = k is a straight line parallel to the x – axis.
6.
An equation of the type y = mx represents a line passing through the
origin, where m is a real number.
7.
Every point on the line satisfies the equation of the line and every
solution of the equation is a point on the line.
8.
The solution of a linear equation is not effected when:
(i) The same number is added or subtracted from both the side of
an equation.
(ii) Multiplying or dividing both the sides of the equation by the
same non zero number.
2
Top Diagrams
1.
Graph of a line passing through the origin.
2.
Graph of a line parallel to x axis.
3
3.
Graph of a line parallel to y axis.
1
IX
Mathematics
Ch 6: Lines and Angles
Chapter Notes
Top Definitions
1.
A line segment is a part of a line which has two end points.
2.
A ray is a part of a line which has only one end point.
3.
A line is a breadth less length which has no end point.
4.
Three or more points when lie on the same line are called collinear points.
5.
Three or more points when don’t lie on a straight line are called non
collinear points.
6.
An angle is formed when two rays originate from the same end point.
7.
The rays making an angle are called the arms of the angle.
8.
The end point from the two rays forming the angle originate is called the
vertex of the angle.
9.
Two angles whose sum is 90° are called complementary angles.
10.
Two angles whose sum is 180° are called supplementary angles.
11.
Two angles are adjacent, if they have a common vertex, a common arm
and their non–common arms are on different sides of the common arm.
12.
If a ray stands on a line, then the sum of the two adjacent angles so formed
is 180° and vice – verse. This property is called as the linear pair axiom.
13.
The vertically opposite angles formed when two lines intersect each other.
There are two pairs of vertically opposite angles.
2
14.
A line which intersects two or more lines at distinct points is called a
transversal.
a
Corresponding angles:
(i)
 1 and  5
(ii)
 2 and  6
(iii)
 4 and  8
(iv)
 3 and  7
(ii)
 3 and  5
(ii)
 2 and  8
b
(i)
c
(i)
Alternate interior angles:
 4 and  6
Alternate exterior angles:
 1 and  7
d Interior angles on the same side of the transversal:
(i)
 4 and  5
(ii)
3 and6
Top Concepts
1.
If a ray stands on a line, then the sum of two adjacent angles so formed is
180°.
3
2.
If the sum of two adjacent angles is 180°, then the non – common arms of
the angles form a line.
3.
If two lines intersect each other, then the vertically opposite angles are
equal.
4.
If a transversal intersects two parallel lines, then
a. Each pair of corresponding angles is equal.
b. Each pair of alternate interior angles is equal.
c. Each pair of interior angles on the same side of the transversal is
supplementary.
5.
If a transversal intersects two lines such that a pair of interior angles on
the same side of the transversal is supplementary, then the two lines are
parallel.
6.
If two lines are parallel to the same line, will they be parallel to each other.
7.
Lines which are parallel to the same line are parallel to each other.
8.
The sum of the angles of a triangle is 180°.
9.
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.
10.
In exterior angle of a triangle is greater then either of its interior opposite
angles.
11.
If a side of a triangle is produced, the exterior angle so formed is equal to
the sum of the two interior opposite angles.
Top Diagrams
1.
A line
4
2.
A ray
3.
A line segment
4.
Intersecting and non intersecting lines.
(i) Intersecting lines
5.
(ii) Non–intersecting (parallel) lines
ABD and DBC are linear pair of angles
5
6.
Types of Angles
1
IX
Mathematics
Chapter 7: Triangles
Chapter Notes
Top Definitions
1.
Two figures are congruent, if they are of the same shape and of the same
size.
2.
Two figures are similar, if they are of the same shape but of different size.
3.
SAS congruence rule: Two triangles are congruent if two sides and the
included angle of one triangle are equal to the two sides and the included
angle of the other triangle.
4.
ASA congruence rule: Two triangles are congruent if two angles and the
included side of one triangle are equal to two angles and the included side
of other triangle.
5.
AAS congruence rule: Two triangles are congruent if any two pairs of angles
and one pair of corresponding sides are equal.
6.
SSS congruent rule: If three sides of one triangle are equal to the three
sides of another triangle, then the two triangles are congruent.
7.
RHS congruence rule: If in two right triangles the hypotenuse and one side
of one triangle are equal to the hypotenuse and one side of the other
triangle, then the two triangles are congruent.
8.
A triangle in which two sides are equal is called an isosceles triangle.
Top Concepts
1.
If two triangles ABC and PQR are congruent under the corresponding A ↔
P, B ↔Q and C ↔ R, then symbolically, it is expressed as
ABC  PQR.
2.
Two circles of the same radii are congruent.
3.
Two squares of the same sides are congruent.
4.
Each angle of an equilateral triangle is of 60°.
5.
In congruent triangles corresponding parts are equal and we write this as
‘CPCT’ for corresponding parts of congruent triangles.
6.
SAS congruence rule holds but not ASS or SSA rule.
7.
Angles opposite to equal sides of an isosceles triangle are equal.
2
8.
The sides opposite to equal angles of a triangle are equal.
9.
RHS stands for Right Angle – Hypotenuse – Side.
10.
If two sides of a triangle are unequal, then the greater angle is opposite to
the greater side.
11.
If two angles of a triangle are unequal, the greater side is opposite to the
greater angle.
12.
The sum of any two sides of a triangle is greater than the third side.
13.
The difference between any two sides of a triangle is less than the third
side.
14.
If the sum of two adjacent angles is 180°, then the non – common arms of
the angles form a line.
Top Diagrams
1.
ABC 
DEF
2.
ABD 
DEF
1
IX
Mathematics
Chapter 8: Quadrilaterals
Chapter Notes
Top Definitions
1.
A quadrilateral is a closed figure obtained by joining four points (with no three
points collinear) in an order.
2.
A diagonal is a line segment obtained on joining the opposite vertices.
3.
Two sides of a quadrilateral having no common end point are called its opposite
sides.
4.
Two angles of a quadrilateral having common arm are called its adjacent
angles.
5.
Two angles of a quadrilateral not having a common arm are called its opposite
angles.
6.
A trapezium is quadrilateral in which one pair of opposite sides are parallel.
7.
In the non – parallel sides of trapezium are equal, it is known as isosceles
trapezium.
8.
A parallelogram is a quadrilateral in which both the pairs of opposite sides are
parallel.
9.
A rectangle is a quadrilateral whose each angle is 90°
10.
A rhombus is quadrilateral whose all the sides are equal.
11.
A square is a quadrilateral whose all sides are equal and each angle is 90°.
12.
A kite is a quadrilateral in which two pairs of adjacent sides are equal.
Top Concepts
1.
Properties of parallelogram:
i
The opposite sides of a parallelogram are parallel.
ii
A diagonal of a parallelogram divides it in two congruent triangles.
iii The opposite sides of a parallelogram are equal.
iv The opposite angles of a parallelogram are equal.
v
The consecutive angles (conjoined angles) of a parallelogram are
supplementary.
2
vi The diagonals of a parallelogram bisect each other.
2.
A diagonal of a parallelogram divides the parallelogram into two congruent
triangles.
3.
In a parallelogram opposite sides are equal.
4.
If each pair of opposite sides of a quadrilateral is equal, then it is a
parallelogram.
5.
In a parallelogram opposite angles are equal.
6.
If in quadrilateral, each pair of opposite angles is equal, then it is a
parallelogram.
7.
The diagonals of a parallelogram bisect each other.
8.
If the diagonals of a quadrilateral bisect other, then it is a parallelogram.
9.
A quadrilateral is a parallelogram, if a pair of opposite sides is equal and
parallel.
10.
Square, rectangle and rhombus are all parallelograms.
11.
Kite and trapezium are not parallelogram.
12.
A square is a rectangle.
13.
A square is a rhombus.
14.
A parallelogram is a trapezium.
15.
Every rectangle is a parallelogram; therefore, it has all the properties of a
parallelogram. Additional properties of a rectangle are:
16.
i
All the (interior) angles of are rectangle are right angles.
ii
The diagonals of a rectangle are equal.
Every rhombus is a parallelogram; therefore, it has all the properties of a
parallelogram. Additional properties of a rhombus are:
i
All the sides of rhombus are equal.
ii
The diagonals of a rhombus intersect at right angles.
iii The diagonals bisect the angles of a rhombus.
17.
Every square is a parallelogram; therefore, it has all the properties of a
parallelogram. Additional properties of a rhombus are:
i
All the sides are equal
ii
All the angles are equal to 90° each
iii Diagonals are equal
3
iv Diagonal bisect each other at right angle
v
Diagonals bisects the angles of vertex
18.
Sum of all the angles of a quadrilateral is 3600.
19.
Mid Point Theorem (Basic Proportionality Theorem): The line segment joining
the mid point of any two sides of a triangle is parallel to the third sides and
equal to half of it.
20.
Converse of mid-point theorem: The line drawn through the mid-point of one
side of a triangle parallel to the another side, bisects the third side.
21.
If there are three or more parallel lines and the interests made by them on a
transversal are equal, then the corresponding intercepts on any other
transversal are also equal.
22.
A quadrilateral formed by joining the mid-points of the sides of a quadrilateral,
in order is a parallelogram.
Top Diagrams
1.
A quadrilateral ABCD.
2.
A trapezium ABCD with sides AB || DC and non parallel sides AD and BC.
3.
A parallelogram ABCD in which AB||DC and AD||BC.
4
a
4.
A rectangle ABCD with AD||BC, AB||DC and A = 90° = B = C = D.
5.
A rhombus ABCD with AB = BD = CD = DA.
6.
A square ABCD in which AB = BC = CD, = DA and A = B = C = D = 90°.
7.
A kite ABCD with AB = AD and BC = CD
5
8.
9.
The relations between special parallelograms can be represented by a Veendiagram.
1
IX
Mathematics
Chapter 9: Area of Parallelograms and Triangles Quadrilaterals
Chapter Notes
Top Definitions
1.
Any side of a parallelogram is called the base.
2.
The length of perpendicular drawn from any point form the parallel sides
to the base is called the (corresponding) altitude or height.
3.
The part of the plane enclosed by a simple closed figure is called a planar
region corresponding to that figure.
4.
The magnitude or measure of that planar region is called its area.
5.
Two figures are called congruent, if they have the same shape and the
same size.
6.
Area of a figure is a number (in same unit) associated with the part of
the plane enclosed by the two properties.
Top Concepts
1.
If two figures A and B are congruent, they must have equal areas.
2.
Two figures having equal areas need not be congruent.
3.
If a planner region formed by a figure T is mad up of two non –
overlapping planner regions formed by figures P and Q, then ar(T) =
ar(P) + ar(Q).
4.
Two figures are said to be on the same base and between the same
parallels, if they have a common base (side) and the vertices (or the
vertex) opposite to the common base of each figure lie on a line parallel
to the base.
5.
Parallelograms on the same base and between the same parallels are
equal in area.
2
6.
Area of a parallelogram is the product of its any side and the
corresponding altitude.
7.
Parallelograms on the same base or equal bases and between the same
parallels are equal in area.
8.
Parallelograms on the same base (or equal bases) and having equal
areas lie between the same parallels.
9.
Two triangles on the same base (or equal base) and between the same
parallel are equal in area.
10.
Area of triangle is half the product of its base (or any side) and the
corresponding altitude (or height).
11.
Two triangles with same base (or equal bases) and equal areas will have
equal corresponding altitudes.
12.
Two triangles having the same base (or equal bases) and equal areas
lie between the same parallels.
13.
Parallelograms on the same base (or equal bases) and having equal
areas lie between the same parallels.
14.
A median of a triangle divides it into triangles of equal areas.
Top Diagrams
1.
Congruent Figures
2.
Parallelograms on the same base and between the same Parallels
3
3.
Triangles on the same base and between the same parallels
1
IX
Mathematics
Chapter 10: Circles
Chapter Notes
Top Definitions
1.
A circle is a collection (set) of all those points in a plane, each one of
which is at a constant distance from a fixed point in the plane.
2.
The fixed point is called the centre and the constant distance is called
the radius of the circle.
3.
All the points lying inside a circle are called its interior points and all
those points which lie outside the circle are called its exterior points.
4.
The collection (set) of all interior points of a circle is called the interior
of the circle while the collection of all exterior points of a circle is called
the exterior of the circle.
5.
A line segment joining two points on a circle is called the chord of the
circle.
6.
A chord passing through the center of the circle is called a diameter of
the circle.
7.
A line which meets a circle in two points is called a secant of the circle.
8.
A polygon is a closed figure made up of three or more line segments
(sides) such that each line segment intersects exactly two others at its
end – points (vertices) and no two line segments which intersect are
collinear.
9.
A polygon is called a regular polygon, if it has all its sides equal and has
all its angles equal.
10.
A (continuous) part of a circle is called an arc of the circle. The arc of a
circle is denoted by the symbol ‘
’.
11.
Circumference: The whole arc of a circle is called the circumference of
the circle.
12.
Semi- circle: One – half of the whole arc of a circle is called a semi –
circle of the circle.
13.
Minor and Major arcs: An arc less than one - half of the whole arc of a
circle is called a minor arc of the circle, and an arc greater than one –
half of the whole arc of a circle is called a major arc of the circle.
14.
Central Angle: Any angle whose vertex is centre of the circle is called a
central angle.
2
15.
Degree measure of an Arc: The degree measure of a minor arc is the
measure of the central angle subtended by the arc.
16.
Congruent Circle: Two circles are said to be congruent if and only if
either of them can be superposed on the other so as to cover it exactly.
17.
Congruent Arc: Two arcs of a circle (or of congruent) circles) are
congruent if either of them can be superposed on the other so as to
cover it exactly.
18.
Sector of a circle: The part of the plane region enclosed by an arc of a
circle and its two bounding radii is called a sector of a circle.
19.
Segment of a circle: A chord of a circle divides it into two parts. Each
part is called a segment.
20.
The part containing the minor arc is called the minor segment, and the
part containing the major arc is called the major segment.
21.
A quadrilateral, all the four vertices of which lie on a circle is called a
cyclic quadrilateral. The four vertices A, B, C and D are said to be
Concyclic points.
Top Concepts
1.
A diameter of circle is its longest chord.
2.
A line can meet a circle at the most in two points.
3.
In a circle, perpendicular from the center to a chord bisects the chord.
4.
In a circle, the line joining the mid – point of a chord to the centre is
perpendicular to the chord.
5.
Equal chords of a circle are equivalent from the centre of the circle.
6.
In a circle, chords which subtend equal angles at the centre are equal.
7.
The two points of intersections determine a chord of the circle.
8.
In a circle, equal chords subtend equal angles at the centre.
9.
In a circle, chords which subtend equal angles at the centre are equal.
10.
Triangle is a polygon with 3 sides.
11.
Quadrilateral is a polygon with 4 sides.
12.
The chords corresponding to congruent arcs are equal.
3
13.
If two arcs of a circle (or of congruent circles) are congruent, then the
corresponding chords are equal.
14.
If two chords of a circle (or of congruent circles) are equal, then their
corresponding arcs (minor, major or semi – circular) are congruent.
15.
One and only one circle can be drawn through three non – collinear
points.
16.
An infinite number of circles can be drawn through a given point P.
17.
An infinite number of circles can be drawn through the two given points.
18.
Perpendicular bisectors of two chords of a circle, intersect each other at
the centre of the circle.
19.
The angle subtended by an arc at the centre is double the angle
subtended by it at any point on the remaining part of the circle.
20.
Angles in the same segment of a circle are equal.
21.
An angle in a semi–circle is a right angle.
22.
The arc of a circle subtending a right angle at any point of the circle in
its alternate segment is a semi–circle.
23.
If a line segment joining two points subtends equal angles at two other
points lying on the same side of the line segment, the four points are
concyclic, i.e., lie on the same circle.
24.
An angle in a semi–circle is a right angle.
25.
The arc of a circle subtending a right angle at any point of the circle in
its alternate segment is a semi–circle.
26.
If a line segment joining two points subtends equal angles at two other
points lying on its same side of the line segment, the four points are
concyclic i.e., lie on the same circle.
27.
If the sum of any pair of opposite angles of a quadrilateral is 180°, then
the quadrilateral is cyclic.
28.
Any exterior angle of a cyclic quadrilateral is equal to the interior
opposite angle.
Top Formulae
1.
Diameter = 2 x Radius.
2.
If the degree measure of AB is θ°, we write m AB is θ°.
4
3.
The degree measure of a semi – circles is 180°
4.
The degree measure of a circle is 360°.
5.
The degree measure of a major arc is (360° - θ°), where θ° is the degree
measure of the corresponding minor arc.
6.
For a quad. ABCD, A + C = 180° or B = D = 180°, then ABCD is
cyclic.
7.
Area of a circle = r2
Top Diagrams
1.
Interior and Exterior of a Circle
2.
Concentric circles
3.
Secant, Diameter and Chord in a circle.
5
4.
5.
6.
7.
Arc of a circle
Circumference of a circle
Semi-Circle
Minor and Major arc
6
8.
Minor and Major Sector
8.
Minor and Major Segment
9.
10.
Circles passing through a point.
Circles passing through two points.
7
11.
12.
Chord bisectors meet at center.
Cyclic Quadrilateral
1
IX
Mathematics
Chapter 11: Geometric Constructions
Chapter Notes
Top Concepts
1.
To construct an angle equal to a given angle.
Given
: Any POQ and a point A.
Required
: To construct an angle at A equal to POQ.
Steps of Construction:
1.
With O as centre and any (suitable) radius, draw an arc to meet
OP at R and OQ at S.
2.
Through A draw a line AB.
3.
Taking A as centre and same radius (as in step 1), draw an arc to
meet AB at D.
4.
Measure the segment RS with compasses.
5.
With d as centre and radius equal to RS, draw an arc to meet the
previous arc at E.
6.
Join AE and produce it to C, then BAC is the required angle equal
to POQ
2.
To bisect a given angle.
Given : Any POQ
Required
: To bisect POQ.
Steps of Construction:
1.
With O as centre and any (suitable) radius, draw an arc to meet
OP at R and OQ at S.
2
2.
With R as centre and any suitable radius (not necessarily) equal
to radius of step 1 (but >
1
2
RS), draw an arc. Also, with S as
centre and same radius draw another arc to meet the previous
arc at T.
3.
3.
Join OT and produce it, then OT is the required bisector of POQ.
To construct angles of 60°, 30°, 120°, 90°, 45°
(i) To construct an angle of 60°
Steps of Construction:
1.
Draw any line OP.
2.
With O as centre and any suitable radius, draw an arc to meet OP
at R.
3.
With R as centre and same radius (as in step 2), draw an arc to
meet the previous arc at S.
4.
Join OS and produce it to Q, then POQ = 60°.
3
(ii) To construct an angle of 30°
Steps of Construction
1.
Construct POQ = 60° (as above).
2.
Bisect POQ (as in construction 2). Let OT be the bisector of
POQ, then POT = 30°
(iii) To construct an angle of 120°
1.
Draw any line OP.
2.
With O as centre and any suitable radius, draw an arc to meet OP
at R.
3.
With R as centre and same radius (as in step 2), draw an arc to
meet the previous arc at T. With T as centre and same radius,
draw another arc to cut the first arc at S.
4.
Join OS and produce it to Q, then POQ = 120°.
4
(iv) To construct an angle of
90° Steps of Construction
1.
Construct POQ = 60°
(as in construction 3(i)).
2.
Construct POV = 120° (as above).
3.
Bisect QOV (as in construction 2). Let OU be the bisector of
QOV, then POU = 90°.
(v) To construct an angle of
45° Steps of Construction
1.
Construct AOP = 90° (as above).
2.
Bisect AOP (as in construction 2).
Let OQ be the bisector of AOP, then AOQ = 45°
5
4.
To bisect a given line segment.
Given
: Any line segment AB.
Required
: To bisect line segment AB.
Steps of Construction:
1.
At A, construct any suitable angle BAC.
2.
At B, construct ABD = BAC on the other side of the line AB.
3.
With A as centre and any suitable radius, draw an arc to meet AC
at E.
4.
From BD, cut off BF = AE.
5.
Join EF to meet AB at G, then EG is a bisector of the line segment
AB and G is mid – point of AB.
(ii) To divided a given line segment in a number of equal part.
6
5.
Divided a line segment AB of length 8 cm into 4 equal part.
Given : A line segment AB of length 8 cm.
Required : To divide line segment 8 cm into 4 equal parts.
Steps of Construction:
1.
Draw lien segment AB = 8 cm.
2.
At A, construct any suitable angle BAX.
3.
At B, construct ABY = BAX on the other side of the line AB.
4.
From AX, cut off 4 equal distances at the points C, D, E and F
such that AC = CD = DE = EF.
5.
With the same radius, cut off 4 equal distances along BY at the
points H, I, J and K such that BH = HI = IJ = JK.
6.
Join AK, CJ, DI, EH and FB. Let CJ, DI and EH meet the line
segment AB at the points M, N and O respectively. Then, M, N
and O are the points of division of AB such that AM = MN = NO =
OB.
6.
To draw a perpendicular bisector of a line segment.
Given
: Any line segment PQ.
Required
: To draw a perpendicular bisector of lien segment PQ.
Steps of Construction:
7
1.
With P as centre and any line suitable radius draw arcs, one on
each side of PQ.
2.
With Q as centre and same radius (as in step 1), draw two more
arcs, one on each side of PQ cutting the previous arcs at A and B.
3.
Join AB to meet PQ at M, then AB bisects PQ at M, and is
perpendicular to PQ, Thus, AB is the required perpendicular
bisector of PQ.
7.
To construct an equilateral triangle when one of its side is given.
E.g.: Construct and equilateral triangle whose each side is 5 cm.
Given
: Each side of an equilateral triangle is 5 cm.
Required
: To construct the equilateral triangle.
Steps of Construction:
1.
Draw any line segment AB = 5 cm.
2.
With A as centre and radius 5 cm draw an arc.
3.
With B as centre and radius 5 cm draw an arc to cut the previous
arc at C.
4.
Join AC and BC. Then ABC is the required triangle.
8
8.
To construct an equilateral triangle when its altitude is given.
E.g.: Construct an equilateral triangle whose altitude is 4 cm.
Steps of Construction:
1.
Draw any line segment PQ.
2.
Take an point D on PQ and At D, construct perpendicular DR to
PQ. From DR, cut off DA = 4 cm.
3.
At A, construct DAS = DAT =
1
2  60 = 30° on either side of
AD. Let AS and AT meet PQ at points B and C respectively. Then,
ABC is the required equilateral triangle.
9.
Construction of a triangle, given its Base, Sum of the other Two sides
and one Base Angle.
9
E.g Construct a triangle with base of length 5 cm, the sum of the other
two sides 7 cm and one base angle of 60°.
Given: In ΔABC, base BC = 5 cm, AB + AC = 7 cm and ABC = 60°
Required : To construct the ΔABC.
Steps of Construction:
1.
Draw BC = 5 cm.
2.
At B, construct CBX = 60°
3.
From BX, cut off BD = 7 cm.
4.
Join CD.
5.
Draw the perpendicular bisector of CD, intersecting BD at a point
A.
6.
10.
Join AC. Then, ABC is the required triangle.
Construction of a triangle, Given its Base, Difference of the Other Two
Sides and one Base Angle.
Eg: Construct a triangle with base of length 7.5 cm, the difference of
the other two sides 2.5 cm, and one base angle of 45°
Given
: In ΔABC, base BC = 7.5 cm, the difference of the other
two sides, AB – AC or AC – AB = 2.5 cm and one base angle is 45°.
Required
: To construct the ΔABC,
CASE (i) AB – AC = 2.5 cm.
Steps of Construction:
10
1.
Draw BC = 7.5 cm.
2.
At B, construct CBX = 45°.
3.
From BX, cut off BD = 2.5 cm.
4.
Join CD.
5.
Draw the perpendicular bisector RS of CD intersecting BX at a
point A.
6.
Join AC. Then, ABC is the required triangle.
CASE (ii) AC – AB = 2.5 cm
Steps of Construction:
1.
Draw BC = 7.5 cm.
2.
At B, construct CBX = 45° and produce XB to form a line XBX’.
3.
From BX’, cut off BD’ = 2.5 cm.
4.
Join CD’.
5.
Draw perpendicular bisector RS of CD’ intersecting BX at a point
A.
6.
Join AC. Then, ABC is the required triangle.
11
e
11.
Construction of a Triangle of Given Perimeter and Base Angles.
Construct a triangle with perimeter 11.8 cm and base angles 60° and
45°.
Given : In ΔABC, AB+BC+CA = 11.8 cm, B = 60° & C = 45°.
Required : To construct the ΔABC.
Steps of Construction:
1.
Draw DE = 11.8 cm.
2.
At D, construct EDP =
DEQ =
1
2
1
of 45° = 22
2 of 60° = 30° and at E, construct
1
2.
3.
Let DP and EQ meet at A.
4.
Draw perpendicular bisector of AD to meet DE at B.
5.
Draw perpendicular bisector of AE to meet DE at C.
6.
Join AB and AC. Then, ABC is the required triangle.
12
1
IX
Mathematics
Chapter 12: Heron’s Formula
Chapter Notes
Top Definitions
1.
The region enclosed with in a simple closed figure is called its area.
2.
A plane figure bounded by four sides is a quadrilateral.
3.
A quadrilateral is a cyclic quadrilateral if all its four vertices lie on the
circumference of the circle.
4.
Semi perimeter is half of the perimeter.
Top Concepts
1.
For every triangle, the values of (s – a), (s – b), and (s – b) are positive.
2.
The line segment joining the mid-point to any of the vertex divides the
triangle in two parts, equal in area.
3.
The diagonal of a quadrilateral divides the quadrilateral into two
triangles.
4.
The diagonal of a parallelogram divides the quadrilateral into two
congruent triangles.
5.
Area of a quadrilateral whose sides and one diagonal are given can be
calculated by dividing the quadrilateral into two triangles and using
Heron’s formula.
Top Formulae
1.
In triangle ABC right angled at B, AB2 + BC2 = AC2
2.
Area of equilateral triangle = 4
length of an equilateral triangle.
3.
Semi-perimeter of equilateral triangle =
4.
Area of a triangle =
3
1
a2 sq units, where ‘a’ is the side
2  base  height
3a
2
2
5.
Area of triangle =
s(s - a)(s - b)(s - c) , s  semi perimeter 
6.
Area of parallelogram = base × height
7.
Area of a triangle =
8.
Area of parallelogram = 2 x (Area of triangle)
9.
Area of cyclic quadrilateral = s(s - a)(s - b)(s - c)(s-d)
a + b + c+d
s  semi perimeter 
2
10.
Area of a rhombus =
11.
Area of a trapezium =
12.
Area of a quadrilateral =
1
2
1
2  base  height
1
2  Pr oduct of diagonals
1
2 height x(sumof parallelsides)
 diagonal x sumof perpendicular fromvertices ondiagonal
a+b+c
2
1
Class IX: Math
Chapter 14: Statistics
Chapter Notes
Top Definitions
1.
Facts or figures collected with a definite purpose are called data.
2.
Statistics deals with collection, presentation, analysis and interpretation
of numerical data.
3.
Arranging data in a order to study their salient features is called
presentation of data.
4.
Data arranged in ascending or descending order is called arrayed data
or an array.
5.
When an investigator with a definite plan or design in mind collects data
first handedly, it is called primary data.
6.
Data when collected by someone else, say an agency or an investigator,
comes to you, is known as the secondary data.
7.
Variable is a quantity that assumes different values.
8.
Range of the data is the difference between the maximum and the
minimum values of the observations.
9.
The small groups obtained on dividing all the observations are called
classes or class intervals and the size is called the class size or class
width.
10.
Class mark of a class is the mid value of the two limits of that class.
11.
A bar graph is the diagram showing a system of connections or
interrelations between two or more things by using bars.
12.
A histogram is the bar graph such that the area over each class interval
is proportional to the relative frequency of data within this interval.
13.
The number of times an observation occurs in the data is called the
frequency of the observation.
14.
A frequency distribution in which the upper limit of one class differs from
the lower limit of the succeeding class is called an Inclusive or
discontinuous Frequency Distribution.
15.
A frequency distribution in which the upper limit of one class coincides
from the lower limit of the succeeding class is called an exclusive or
continuous Frequency Distribution.
2
16.
A bar graph is a pictorial representation of data in which rectangular
bars of uniform width are drawn with equal spacing between them on
one axis, usually the x axis. The value of the variable is shown on the
other axis that is the y axis.
17.
A histogram is a set of adjacent rectangles whose areas are proportional
to the frequencies of a given continuous frequency distribution.
18.
The Cumulative Frequency of a class-interval is the sum of frequencies
of that class and the classes which precede (come before) it.
19.
The mean value of a variable is defined as the sum of all the values of
the variable divided by the number of values.
20.
Median is the value of middle most observation(s).
21.
Mode of a statistical data is the value of that variate which has the
maximum frequency.
Top Concepts
1.
In case of continuous frequency distribution, the upper limit of a class
is not to be included in that class while in discontinuous both the limits
are included.
2.
The height of rectangles corresponds to the numerical value of the data.
3.
Frequency polygons are a graphical device for understanding the shapes
of distributions.
4.
Bar charts are used for comparing two or more values.
5.
A histogram differs from a bar chart, as in the former it is the area of
the bar that denotes the value, not the height.
6.
The height of the rectangle as the ratio of the frequency of the class to
the width or size of the class.
7.
Last cumulative frequency is always the sum total of all the frequencies.
8.
If both a histogram and a frequency polygon are to be drawn on the
same graph, then we should first draw the histogram and then join the
mid-points of the tops of the adjacent rectangles in the histogram with
line-segments to get the frequency polygon.
9.
If classes are not of equal width, then the height of the rectangle is
calculated by the ratio of the frequency of that class, to the width of that
class.
3
10.
A measure of central tendency tries to estimate the central value which
represents the entire data.
11.
The three measures of central tendency for ungrouped data are mean,
mode and median.
12.
The disadvantage of arithmetic mean is that it is affected by extreme
values.
13.
The median is to be calculated only after arranging the data in ascending
order or descending order.
22.
Average height is the modal value.
23.
Disadvantage of the mode is that it is not uniquely defined in many
cases.
24.
The data is symmetric about the mean position when the three averages
mean median and mode are all equal.
25.
The data is asymmetric when the three measures are unequal.
14.
The variate corresponding to the highest frequency is to be taken as
the mode and not the frequency.
Top Formulae
Range
1. Class size = Number of classes
2. Class size = Upper limit – Lower Limit
3.
Mean ( x ) 
1
n
n
 xi
i1
4.
Mean ( x) 
 f i xi
f

i
5. (i) If number of observations (n) is odd, Median = ( n 1 )th observation
2
 n th n

 ( 2 )  ( 2 1)th 
(ii) If n is even, then median 
 observation
2




4
Top Diagrams
1.
Symmetric Distribution
2.
Asymmetrical or skewed distribution
5
3.
Bar Graph
4.
Mean < Mode
6
5.
Mode < Mean
6.
Frequency Polygons
7
7.
A histogram