Topics
Number System
Rational Numbers
- Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
- Rational Numbers on a Number Line
- Inserting Rational Numbers Between Two Given Rational Numbers
- Method of Finding a Large Number of Rational Numbers Between Two Given Rational Numbers
Exponents
- Concept of Exponents
- Law of Exponents (For Integral Powers)
- Negative Integral Exponents
- More About Exponents
Squares and Square Root
- Concept of Square Roots
- Finding Square Root by Division Method
- Finding Square Root Through Prime Factorisation
- To Find the Square Root of a Number Which is Not a Perfect Square (Using Division Method)
- Properties of Square Numbers
Cubes and Cube Roots
Playing with Numbers
- Arranging the Objects in Rows and Columns
- Generalized Form of Numbers
- Some Interesting Properties
- Letters for Digits (Cryptarithms)
- Divisibility by 10
- Divisibility by 2
- Divisibility by 5
- Divisibility by 3
- Divisibility by 6
- Divisibility by 11
- Divisibility by 4
Sets
- Concept of Sets
- Representation of a Set
- Cardinality of a Set
- Types of Sets
- Subset
- Proper Subset
- Super Set
- Universal Set
- Complement of a Set
- Difference of Two Sets
- Distributive Laws
- Venn Diagrams
Ratio and Proportion
Percent and Percentage
Profit, Loss and Discount
- Overhead Expenses
- To Find S.P., When C.P. and Gain (Or Loss) Percent Are Given
- To Find C.P., When S.P. and Gain (Or Loss) Percent Are Given
- Concept of Discount
Interest
- Calculation of Interest
- Concept of Compound Interest
- To Find the Principle (P); the Rate Percent (R) and the Time
- Interest Compounded Half Yearly
- Applications of Compound Interest Formula
Direct and Inverse Variations
- Types of Variation
- Unitary Method
- Concept of Arrow Method
- Time and Work
Algebra
Algebraic Expressions
- Algebraic Expressions
- Degree of Polynomial
- Factors and Common Factors
- Classification of Terms in Algebra
- Combining like Terms
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Polynomial
- Multiplying a Polynomial by a Polynomial
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Simplification of Expressions
Identities
- Algebraic Identities
- Product of Sum and Difference of Two Terms
- Expansion Form of Numbers
- Important Formula of Expansion
- Cubes of Binomials
- Application of Formulae
Factorisation
- Factorisation by Taking Out Common Factors
- Factorisation by Taking Out Common Factors
- Factorisation by Grouping
- Factorisation by Difference of Two Squares
- Factorisation of Trinomials
- Factorising a Perfect Square Trinomial
- Factorising Completely
Linear Equations in One Variable
Linear Inequations
- Pair of Linear Equations in Two Variables
- Replacement Set and Solution Set
- Operation of Whole Numbers on Number Line
- Concept of Properties
Geometry
Understanding Shapes
Special Types of Quadrilaterals
Constructions
- Introduction of Constructions
- Construction of an Angle
- To Construct an Angle Equal to Given Angle
- To Draw the Bisector of a Given Angle
- Construction of an Angle Bisector: 30°, 45°, 60°, 90°
- Construction of Bisector of a Line
- The Perpendicular Bisector
- Construction of Parallel Lines
- Constructing a Quadrilateral
- Construction of Parallelograms
- Construction of a Rectangle When Its Length and Breadth Are Given.
- Construction of Rhombus
- Square: Properties and Construction
- Reflection Symmetry (Mirror Symmetry)
Representing 3-D in 2-D
- 2dimensional Perspective of 3dimensional Objects
- Concept of Polyhedron
- Faces, Edges and Vertices of Polyhedron
- Euler's Formula
- Concept of Polyhedron
- Nets of 3D Figures
Mensuration
Area of a Trapezium and a Polygon
Surface Area, Volume and Capacity
Data Handling (Statistics)
Data Handling
- Mathematical Data Collection and Organisation
- Frequency
- Raw Data, Arrayed Data and Frequency Distribution
- Cumulative Frequency and Cumulative Frequency Table
- Frequency Distribution Table
- Class Intervals and Class Limits
- Frequency Distribution and Its Applications
Probability
- Definition: Perpendicular Bisector
- Properties
- Construction of the Perpendicular Bisector
- Activity: Understanding the Perpendicular Bisector
Definition: Perpendicular Bisector
A perpendicular bisector is defined as a line or a line segment that divides a given line segment into two parts of equal length and forms a right angle (90°) with it.
Properties
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Uniqueness: Every line segment has precisely one unique perpendicular bisector.
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The intersection point (M) of the bisector and the segment is always the segment's midpoint, irrespective of the method of construction.
Construction of the Perpendicular Bisector
1. Draw a line segment and name it AB.
2. Place the compass at point A,
- Take a distance more than half the length of AB.
- Draw two arcs — one above and one below the segment AB.
3. Now place the compass at point B,
- Use the same compass width.
- Draw two more arcs to intersect the previous arcs.
- Mark the points of intersection as P (above) and Q (below).
4. Create a straight line connecting points P and Q.
5. This line PQ is the perpendicular bisector of segment AB.
6. Verify:
- Measure the lengths of AP and PB → they should be equal.
- Measure the angle between line PQ and segment AB → it should be 90°.
Activity: Understanding the Perpendicular Bisector
Steps:
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Draw a line segment AB.
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Mark its midpoint M using a compass or by measurement.
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Draw two lines, p and q, passing through the midpoint M.
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Ensure both lines divide AB into two equal parts — making them bisectors.
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Use a protractor to measure the angle between each line (p and q) and segment AB.
Observation:
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The angle between line p and segment AB is 90°, a right angle.
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Line p passes through the midpoint M and forms a right angle with AB, so it is a perpendicular bisector.
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Line q, although passing through M, does not form a 90° angle with AB. Hence, q is only a bisector, not a perpendicular bisector.
Conclusion:
A line is called a perpendicular bisector only if:
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It passes through the midpoint of the given segment, and
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It forms a right angle (90°) with the segment.
If a line only divides the segment into two equal parts without forming a 90° angle, it is simply a bisector, not a perpendicular bisector.
