Mathematics Class 9 ICSE CISCE Topics and Syllabus

• Conjugate Surds

• Absolute value

• Simple Interest for one year
• Simple Interest for multiple years

( a + b )² = a² + 2ab + b² .

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 )

• ( 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

• ( a x b )^m = a^m x b^m and `(a/b)^m = a^m/b^m`
  
  
• If a ≠ 0 and n is a positive integer, then `nsqrta` = a^1/n 
  
  
• `a^(m/n) = nsqrt (a^m) ; Where a ≠ 0. `
  
  
• For any non - zero number a,
  
  `a^n = 1/( a^-n ) and a^(-n) = 1/(a^n)`  
  
  
• Any non - zero number raised to the power zero is always equal to unity ( i.e., 1)

• 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.

• 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. 

• If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.

• 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

• Rectilinear means along a straight line or in a straight line or forming a straight line.
• A plane figure bounded by straight lines is called a rectilinear figure.
• A closed plane figure, bounded by at least three line segments, is called a polygon.

• A Polygon is named by the number of sides in it,
  

  No. of sides	3	4	5	6	7	8
  Name	Triangle	Quadrilateral	Pentagon	Hexagon	Heptagon	Octagon

• Concave Polygon : If at least one angle of a polygon is greater than 180°, the polygon is called a convex polygon.
• Convex Polygon : If each angle of a polygon is less than 180°, it is called a concave polygon.

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.

1. To construct a trapezium ABCD, whose four sides are given.

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°`

• 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.

1. Continuous Variable
2. Discrete Variable
3. Raw and Arrayed Data

1. Ungrouped Frequency Distribution Table:
2. Grouped Frequency Distribution Table:
   - Inclusive Frequency Distribution
   - Exclusive Frequency Distribution

• Less than Cumulative frequency less than the upper class limit
• Cumulative frequency more than or equal to the lower class limit

• Bar graph
• Pie graph
• Histogram

1. Histogram
2. Frequency Polygon

• Right Circular Cylinder
• Hollow Cylinder

• Cost = Rate x Quantity

• Volume = Area of cross - section x length
• Surface area ( excluding cross - section ) = Perimeter of cross - section x length

• The volume of water that flows in unit time = Area of cross-section x speed of flow of water.

• Graph
• Graph sheets

• Similar triangles
• Criteria of Similarity
  - AA Criterion of similarity
  - SAS Criterion of similarity
  - SSS Criterion of similarity
• Construction of similar triangles