# Chapter 4 – Geometry

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

In the ICSE syllabus of class 9th mathematics, the topic ‘Geometry‘ is an essential chapter as it forms the basis of higher-level mathematics. It covers important topics related to geometry and shapes. It is also used in successive chapters of class 9th mathematics, so a clear understanding of geometry will help you learn other mathematical concepts and chapters better.

The chapter covers the following topics:

• Introduction to geometry
• Triangles and types of triangles
• Pythagoras theorem
• Rectilinear figures
• Circle

#### Introduction to Geometry

Geometry is an important branch of mathematics that involves studying various shapes, figures, dimensions, angles, etc. It provides formulas to calculate the area, perimeter, circumference, and volume of various figures. Geometrical figures are divided into two types based on dimensions:

1. 2D figures: Shapes with only 2 dimensions, length and breadth, are called 2D figures. We can calculate the area and perimeter of 2D figures. Example: triangles, circles, etc.
2. 3D figures: Shapes with 3 dimensions, length, breadth, and height, are called 3D figures. We can calculate the surface area and volume of 3D figures. Example: cones, cubes, cylinders, etc.

#### Triangles and Types of Triangles

Triangles are three-sided 2D figures that satisfy the following conditions:

• The sum of any two sides is greater than the third side of the figure.
• The angle exactly opposite to the longest side is the largest angle of the figure.
• The total sum of the interior angles of the figure is exactly 180°

Area of triangle = ½ × (b × h), where b is base and h is the height of the triangle

The perimeter of a triangle = Sum of all sides

Types of Triangles

Triangles are differentiated based on their angles and sides.

Based on the angles:

• Acute triangle: Each angle of the triangle is less than 90°.
• Right-angled triangle: One angle of the triangle is equal to 90°.
• Obtuse triangle: One angle of the triangle is greater than 90° but less than 180°.
• Equiangular triangle: Each angle of the triangle is equal to 60°.

Based on the sides:

• Scalene triangle: The lengths of all sides are different.
• Isosceles triangle: The lengths of any two sides are equal.
• Equilateral triangle: All sides of the triangle are equal.

Congruent Triangles

Two triangles are said to be congruent if both are of the same shape and size. For example, two equilateral triangles of side 6 cm are congruent to each other.

Similar Triangles

Two triangles are said to be similar if both are of the same shape but different sizes. For example, two right-angled triangles are similar to each other but might not be congruent.

#### Proportionality Theorem

Also known as Thales’s theorem, it states that if a line DE is drawn parallel to the base BC of triangle ABC, then AD/DB = AE/EC.

#### Pythagoras Theorem

This theorem applies only to right-angled triangles where one angle is equal to 90°. The longest side of the right-angled triangle is opposite the 90° angle and is known as hypotenuse.

According to this theorem, the square of the hypotenuse of a right-angled triangle is always equal to the sum of the squares of the other two sides of the triangle.

Example: In a right-angled triangle ABC, where AB is the hypotenuse, AB^2 = BC^2 + AC^2.

#### Rectilinear Figures

Closed geometrical figures made of straight lines are known as rectilinear figures. In layman’s terms, closed figures made of at least three line segments are known as polygons or rectilinear figures.

Polygons are classified based on the sides:

• 3-sided figures: Triangles
• 4-sided figures: Quadrilateral
• 5-sided figures: Pentagon
• 6-sided figures: Hexagon
• 7-sided figures: Heptagon
• 8-sided figures: Octagon

A regular polygon is a figure where all sides, exterior angles, and interior angles are equal.

#### Formulas for Rectilinear Figures or Polygons

The following are the formulas for polygons with side ‘n’:

• The sum of interior angles is equal to (2n-4) × 90°
• The sum of all the exterior angles of the polygon is equal to 360°
• The exterior angle of a regular polygon is equal to 360°/n, whereas an interior angle of a regular polygon is equal to (2n-4) × 90°/n.
• The sum of an exterior angle and interior angle is 180° at every vertex of the triangle.

#### Circle

Circles are 2D figures measured on the basis of the radius. As it is a 2D shape, it has both area and perimeter.

Area = πr^2

Circumference (Perimeter) = 2πr

Properties of Circle

• The angle of a semicircle is 90°.
• The angles from a chord in the same segment are equal.
• The central angle of the circle is twice the angle on the circumference of the circle.
• The perpendicular line drawn from the centre of the circle divides the chord into two equal parts.

#### Conclusion

This chapter is an important topic for class 9th mathematics and an essential concept for advanced and higher-level mathematics. Sound knowledge of this chapter will prepare you for advanced lessons.

#### FAQs

1. What do you mean by geometry?
Geometry is a branch of mathematics that involves studying various shapes, dimensions, angles, etc.
2. What is the formula to calculate the sum of interior angles of a polygon?
The sum of interior angles is equal to (2n-4) × 90°
3. What is a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral inside a circle.
4. What is the property of a cyclic quadrilateral?
The sum of the opposite angles of a cyclic quadrilateral is 180°.
5. Are all congruent triangles similar?
Yes, all congruent triangles are similar, but the opposite is not true.

Geometry is an exciting chapter of mathematics. The visual representation of its concepts is essential. The MSVgo app is a video library platform that helps you learn topics through visuals and practical models. Make sure to check out the explanatory videos at MSVgo for a better understanding of these concepts.

### High School Physics

• Alternating Current
• Atoms
• Communication Systems
• Current Electricity
• Dual nature of Radiation and Matter
• Electric Charges and Fields
• Electricity
• Electromagnetic Induction
• Electromagnetic Waves
• Electron Beams and Radioactivity
• Electrons and Photons
• Electrostatic Potential and Capacitance
• Fluid Pressure
• Force and Acceleration
• Force And Laws Of Motion
• Gravitation
• Internal Energy
• Kinetic Theory
• Law of motion
• Light – Reflection And Refraction
• Magnetic Effects Of Electric Current
• Magnetism and Matter
• Management Of Natural Resources
• Mechanical properties of Fluids
• Mechanical properties of Solids
• Motion
• Motion in a plane
• Motion in a straight line
• Moving Charges and Magnetism
• Nuclear Energy
• Nuclei
• Oscillations
• Our Environment
• Paths of Heat
• Physical world
• Ray optics and optical instruments
• Semiconductor Devices
• Semiconductor Electronics: Materials, Devices and Simple Circuits
• Simple Machines
• Sound
• Sources Of Energy
• Specific and Latent Heats
• Spherical Mirrors
• Static Electricity
• Systems of Particles and Rotational motion
• Thermal properties of matter
• Thermodynamics
• Units and Measurement
• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
• Aliphatic and Aromatic Hydrocarbons
• Alkyl and Aryl Halides
• Amines
• Analytical Chemistry
• Atomic Structure
• Atoms And Molecules
• Basic concepts of Chemistry
• Biomolecules
• Carbon And Its Compounds
• Carboxylic acids and Acid Derivatives
• Chemical Bonding and Molecular Structures
• Chemical Energetics
• Chemical Equilibria
• Chemical Kinetics
• Chemical Reactions And Equations
• Chemical Reactions and Their Mechanisms
• Chemistry in Everyday Life
• Chemistry of p-Block elements
• Chemistry of Transition and Inner Transition
• Classification of Elements
• Coordination Compounds
• Cyanide, Isocyanide, Nitro compounds and Amines
• Electrochemistry
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• Elements, Compounds and Mixtures
• Environmental Chemistry
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• Matter
• Matter Around Us
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• Periodic Classification of Elements
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• Preparation, Properties and Uses of Compounds
• Principles and Processes of Isolation of Elements
• Redox Reactions
• Relative Molecular Mass and Mole
• States of Matter
• Structure Of The Atom
• Study of Compounds
• Study of Gas Laws
• Study of Representative Elements
• Surface Chemistry
• The d-block and f-block elements
• The Gaseous State
• The p-Block Elements
• The Periodic Table
• The s-Block Elements
• The Solid State
• Thermodynamics

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• Absorption and Movement of Water in Plants
• Adolescent Issues
• Anatomy of Flowering Plants
• Animal Kingdom
• Bacteria and Fungi-Friends and Foe
• Biodiversity and Conservation
• Biofertilizers
• Biological Classification
• Biomedical Engineering
• Biomolecules
• Biotechnology and its Applications
• Biotic Community
• Body Fluids and Circulation
• Breathing and Exchange of Gases
• Cell – Unit of Life
• Cell Cycle and Cell Division
• Cell Division and Structure of Chromosomes
• Cell Reproduction
• Cellular Respiration
• Chemical Coordination and Integration
• Circulation
• Control And Coordination
• Crop Improvement
• Digestion and Absorption
• Diversity In Living Organisms
• Ecosystem
• Environmental Issues
• Excretory Products and their Elimination
• Flowering Plants
• Genes and Chromosomes
• Health and Diseases
• Health and Its Significance
• Heredity And Evolution
• Heredity and Variation
• How Do Organisms Reproduce?
• Human Diseases
• Human Eye And Colourful World
• Human Health and Disease
• Human Population
• Human Reproduction
• Hygiene
• Improvement In Food Resources
• Integumentary System- Skin
• Kingdom Fungi
• Kingdom Monera
• Kingdom Protista
• Life Processes
• Locomotion and Movement
• Microbes in Human Welfare
• Mineral Nutrition
• Molecular Basis of Inheritance
• Morphology of Flowering Plants
• Neural Control And Coordination
• Nutrition in Human Beings
• Organism and Population
• Photosynthesis
• Photosynthesis in Higher Plants
• Plant Growth and Development
• Plant Kingdom
• Pollination and Fertilization
• Pollution; Sources and its effects
• Principles of Inheritance and Variation
• Reproduction and Development in Angiosperms
• Reproduction in Organisms
• Reproductive Health
• Respiration in Human Beings
• Respiration in Plants
• Respiratory System
• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
• Transport in Plants

### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Algebra – Quadratic Equations
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Quadrilaterals
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
• Number Systems
• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
• Trignometry – Height and Distance
• Trignometry – Identities
• Trignometry – Introduction

### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Chemistry in Your Life
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
• Electric Current And Its Effects
• Electricity And Circuits
• Elements and Compounds
• Fibre To Fabric
• Food production and management
• Force And Pressure
• Forests: Our Lifeline
• Friction
• Fun With Magnets
• Garbage In, Garbage Out
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• Health and Hygiene
• Heat
• Hydrogen
• Life Processes: Nutrition in Animals and Plants
• Light, Shadows And Reflections
• Materials: Metals And Non-Metals
• Matter and Its States
• Metals and Non-metals
• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
• Nutrition In Animals
• Nutrition In Plants
• Organization in Living Things
• Our Environment
• Physical And Chemical Changes
• Pollution and conservation
• Pollution Of Air And Water
• Reaching The Age Of Adolescence
• Reproduction In Animals
• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
• Simple Machines
• Soil
• Some Natural Phenomena
• Sorting Materials Into Groups
• Sound
• Stars And The Solar System
• Structure of Atom
• Synthetic Fibers And Plastics
• The Living Organisms And Their Surroundings
• Transfer of Heat
• Transformation of Substances
• Transportation In Animals And Plants
• Universe
• Waste-water Story
• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• Addition
• Area and Its Boundary
• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time
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