# Mathematics Class 9 ICSE CISCE Topics and Syllabus

## Syllabus

2 Compound Interest [Without Using Formula]
4 Expansions
• Algebraic Identities

( a + b )2 = a2 + 2ab + b2 .

• Expansion of (A + B)3
• ( a + b ) ( a - b ) = a2 - b2

• ( a + b )3 = a3 + 3a2b + 3ab2 + b3

• ( a - b )3 = a3 - 3a2b + 3ab2 - b3

• a3 + b3 = ( a + b )3 - 3ab ( a + b )

• a3 - b3 = ( a - b )3 + 3ab ( a - b )

• ( a + 1/a )3 = a3 + 1/(a^3) + 3 ( a + 1/a )

• ( a - 1/a )3 = a3 - 1/(a^3) - 3 ( a - 1/a )

• a3 + 1/(a^3) = ( a + 1/a )3 - 3 ( a + 1/a )

• a3 - 1/(a^3) = ( a - 1/a )3 - 3 ( a - 1/a )

• a3 + b3 + c3 = 3abc
• Expansion of Formula

1.  Expansion of ( x + a ) ( x + b ) :

• ( x + a ) ( x + b ) = x2 + ( a + b ) x + ab

• ( x + a ) ( x - b ) = x2 + ( a - b ) x - ab

• ( x - a ) ( x + b ) = x2 - ( a - b ) x - ab

• ( x - a ) ( x - b ) = x2 - ( a + b ) x + ab

2. Expansion of ( a + b + c )2 :

• ( a + b + c )2 = a2 + b2 + c2 + 2 ( ab + bc + ca )

• ( a + b - c )2 = a2 + b2 + c2 + 2 ( ab - bc - ca )

• ( a - b + c )2 = a2 + b2 + c2 - 2 ( ab + bc - ca )

• ( a - b - c )2 = a2 + b2 + c2 - 2 ( ab - bc + ca )
• Special Product
• ( x + a ) ( x + b ) ( x + c ) = x3 + ( a + b + c ) x3 + ( ab + bc + ca ) x + abc

• ( a + b ) ( a2 - ab + b2 ) = a3 + b3

• ( a - b ) ( a2 + ab + b2 ) = a3 - b3

• ( a + b + c ) ( a2 + b2 + c2 - ab - bc - ca ) = a3 + b3 + c3 - 3abc
• Elimination Method of Solving Simultaneous Equations
7 Indices [Exponents]
9 Triangles [Congruency in Triangles]
• Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
• Relation Between Sides and Angles of Triangle
• If all the sides of a triangle are of  different lengths, its angles are also of different measures in such a way that, the greater side has greater angle opposite to it.

• If all the angles of a triangle have different measures, its sides are also of different lengths in such a way that, the greater angle has greater side opposite to it.

• If any two sides of a triangle are equal, the angles opposite to them are also equal. Conversely, if any two angles of a triangle are equal, the sides opposite to them are also equal.

• If all the sides of a triangle are equal, all its angles are also equal. Conversely, if all the angles of a triangle are equal, all its sides are also equal.
• Important Terms of Triangle
• Median : The median of a triangle, corresponding to any side, is the line joining the mid-point of that side with the opposite vertex.

• Centroid : The point of intersection of the medians is called the centroid of the triangle.

• Altitude : An altitude of a triangle, corresponding to any side, is the length of the perpendicular drawn from the opposite vertex to that side.

• Orthocentre : The point of intersection of the altitudes of a triangle is called the orthocentre.

• Corollary 1 : If one side of a triangle is produced, the exterior angle so formed is greater than each of the interior opposite angles.

• Corollary 2 : A triangle cannot have more than one right angle.

• Corollary 3 : A triangle cannot have more than one obtuse angle.

• Corollary 4 : In a right angled triangle, the sum of the other two angles ( acute angles ) is 90°.

• Corollary 5 : In every triangle, at least two angles are acute.

• Corollary 6 : If two angles of a traingle are equal to two angles of any other triangle, each to each, then the third angles of both the triangles are also equal.
• Concept of Congruency of Triangles
• Two triangles are said to be congruent to each other, if on placing one over the other, they exactly coincide.
• In fact, Two triangle are congruent, If all the angles and all the sides of one triangle are equal to the corresponding angles and the corresponding sides of the other triangle each to each.

• The symbol ≅ is read as "is congruent to"
• In Congruent triangles, the sides and the angles that coincide by superposition are called corresponding sides and corresponding angles.
•  Corresponding parts of Congruent Triangles are also Congruent.
Abbreviated as : C.P.C.T.C.
• Condition for Congruency of Triangle
1. If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, the triangles are congruent.
Abbreviated as : S.A.S.
2. If  two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, the triangles are congruent.
Abbreviated as : A.S.A.
3. If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, the triangles are congruent.
Abbreviated as : A.A.S.
4. If three sides of one triangle are equal to three sides of the other triangle, each to each, the triangles are congruent.
Abbreviated as : S.S.S.
5. Two right-angled triangles are congruent, if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of the other triangle.
Abbreviated as : R.H.S.
11 Inequalities
12 Mid-point and Its Converse [ Including Intercept Theorem]
• The Mid-point Theorem
• Theorem of midpoints of two sides of a triangle : The line segment joining the mid-points of any two sides of a triangle is parallel to the third side, and is equal to half of it.

• Converse of midpoint theroem : The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side.
• Equal Intercept Theorem
• If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.
13 Pythagoras Theorem [Proof and Simple Applications with Converse]
• Pythagoras Theorem
• Regular Polygon
• If all the sides and all the angles of a polygon are equal, it is called a regular polygon.
• Sum of interior angles of an 'n' sided polygon ( whether it is regular or not) = ( 2n - 4 )rt. angles and sum of its exterior angles = 4 right angles = 360°
• At each vertex of every polygon, Exterior angle + Interior angle = 180°.
• Each interior angle of a regular polygon = [( 2n - 4 ) "rt. angles"]/[n] = [( 2n - 4 ) xx 90°]/n
• Each exterior
14 Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
15 Construction of Polygons (Using Ruler and Compass Only)
• Construction of an Angle
1. To construct a quadrilateral, whose four sides and one angle are given.
2. To construct a quadrilateral, whose three sides and two consecutive angles are given.
3. To construct a quadrilateral, whose four sides and one diagonal are given.
4. To construct a quadrilateral, whose three sides and two diagonals are given.
• Construction of Parallelograms
1. To construct a parallelogram, whose two consecutive sides and the included angle are given.
2. To construct a parallelogram, whose one side and both the diagonals are given.
3. To construct a parallelogram, whose two consecutive sides and one diagonal are given.
4. To construct a parallelogram, whose two diagonal and included angle are given.
5. To construct a parallelogram, whose two adjacent sides and height are given.
• Construction of Trapezium
1. To construct a trapezium ABCD, whose four sides are given.
• Construction of Rectangles
1. To construct a rectangle whose adjacent sides are given.
2. To construct a rectangle, whose one side and one diagonal are given.
• To Construct a Regular Hexagon

Method 1 : Each interior angle of a regular hexagon is 120° and its opposite sides are parallel.

Method 2 :
The length of the side of a regular hexagon is equal to the radius of its circumcircle.

Method 3 : The angle subtended by each side of a regular hexagon at the centre of its circumcircle is (360°)/6 = 60°

16 Area Theorems [Proof and Use]
• Introduction of Area Theorems
• Area of a triangle = 1/2 x base x height
• Area of a rectangle = length x breadth
• Area of a parallelogram = base x height
• Figures Between the Same Parallels
• Parallelograms on the same base and between the same parallels are equal in area.
• Corollary : The area of a parallelogram is equal to the area of a rectangle on the same base and between the same parallels.
• The area of a triangle is half that of a parallelogram on the same base and between the same parallels.
• Triangles on the same base and between the same parallels are equal in area.
• Corollaries :
1.
Parallelograms on equal bases and between the same parallels are equal in area.
2. Area of a triangle is half the area of the parallelogram if both are on equal bases and between the same parallels.
3. Two triangles are equal in area if they are on the equal bases and between the same parallels.
• Triangles with the Same Vertex and Bases Along the Same Line
18 Statistics
19 Mean and Median (For Ungrouped Data Only)
20 Area and Perimeter of Plane Figures
21 Solids [Surface Area and Volume of 3-d Solids]
22 Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and Their Reciprocals]
23 Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]
24 Solution of Right Triangles [Simple 2-d Problems Involving One Right-angled Triangle]
27 Graphical Solution (Solution of Simultaneous Linear Equations, Graphically)
28 Distance Formula
30 Construction of Triangles
31 Changing the Subject of a Formula