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Chapter 1 - Number System. Chapter 2 - Polynomials. Chapter 3 - Coordinate Geometry. Chapter 4 - Linear Equations in Two Variables. Chapter 5 - Introductions to Euclids Geometry. Chapter 6 - Lines and Angles. Chapter 7 - Triangles. Chapter 8 - Quadrilaterals. Chapter 9 - Areas of Parallelogram and Triangles.
Chapter 10 - Circles. Chapter 11 - Constructions. Chapter 12 - Herons formula. Chapter 13 - Surface area and Volumes. Chapter 14 - Statistics. Chapter 15 - Probability. Chapter 5 Euclid's Geometry Class 9 is divided into five sections and four exercises. The first section is the introduction with no exercise. The Second and Third section explains the two important properties whereas the Fourth and Fifth sections revisit the topics taught in class 9. Exercise 5. Geometry appears to have originated from the need for measuring land.
Euclidean geometry is the study of plane and solid figures on the basis of axioms and postulates employed by the Greek mathematician named Euclid. It deals with the properties and relationship between all the things. The studies of space and solids were summarised into statements called definitions. The study of solids led to the study of surfaces, curves, lines and points.
A solid object has shape, size and position. A solid always has a surface which separates one part from the other. The boundaries of these surfaces are either curves or straight lines. And these straight lines are made up of points. All these gave rise to the concept of dimensions:. Solids - Three Dimensional. Surface - Two Dimensional.
Point - No Dimensions. Given below are the few important points mentioned in the book:. A point is that which has no part.
A line is a breadthless length. The ends of a line are points. A straight line is a line which lies evenly with the points on itself. A surface is that which has length and breadth only.
The edges of a surface are lines. A plane surface is a surface which lies evenly with the straight lines on itself. These were considered as universal truth and were divided into two parts: Axioms common to entire mathematics and Postulates assumptions specific to geometry. Elements is a collection of definitions, postulates axioms , propositions theorems and constructions , and mathematical proofs of the propositions.
Things are equal to one another if those things are equal to the same thing. The wholes are equal if equals are added to equals. The remainders are equal if equals are subtracted from equals. Things are equal to one another if they coincide with one another.
The whole will be greater than the part. It can be seen that the definition of a few terms needs extra specification. Now let us discuss these Postulates in detail. This postulate states that at least one straight line passes through two distinct points but he did not mention that there cannot be more than one such line.
Although throughout his work he has assumed there exists only a unique line passing through two points. In simple words what we call a line segment was defined as a terminated line by Euclid. Therefore this postulate means that we can extend a terminated line or a line segment in either direction to form a line. In the figure given below, the line segment AB can be extended as shown to form a line.
Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment. All the right angles i. Need for Geometry in Ancient Civilizations. Geometry was studied in various forms in every ancient civilisation, be it in Egypt, Babylonia, China, India, Greece, the Incas, etc. The people of these civilisations faced several practical problems which required the development of geometry in various ways.
For example, whenever the river Nile overflowed, it wiped out the boundaries between the adjoining fields of different landowners. After such flooding, these boundaries had to be redrawn. For this purpose, the Egyptians developed a number of geometric techniques and rules for calculating simple areas and also for doing simple constructions. The knowledge of geometry was also used by them for computing volumes of granaries, and for constructing canals and pyramids.
They also knew the correct formula to find the volume of a truncated pyramid. Pyramid, Harappa And Mohenjo - Daro. You know that a pyramid is a solid figure, the base of which is a triangle, or square, or some other polygon, and its side faces are triangles converging to a point at the top.
In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc. It was a highly organised society.
The cities were highly developed and very well planned. For example, the roads were parallel to each other and there was an underground drainage system. The houses had many rooms of different types. This shows that the town dwellers were skilled in mensuration and practical arithmetic. The bricks used for constructions were kiln fired and the ratio length: breadth: thickness, of the bricks, was found to be 4: 2: 1.
Other Ancient Uses of Geometry. The geometry of the Vedic period originated with the construction of altars or vedis and fireplaces for performing Vedic rites.
The location of the sacred fires had to be in accordance with the clearly laid down instructions about their shapes and areas if they were to be effective instruments. Square and circular altars were used for household rituals, while altars whose shapes were combinations of rectangles, triangles and trapeziums were required for public worship.
The Sri Yantra given in the Atharvaveda consists of nine interwoven isosceles triangles. These triangles are arranged in such a way that they produce 43 subsidiary triangles. Though accurate geometric methods were used for the constructions of altars, the principles behind them were not discussed.
Documents and Proofs of Geometry. These examples show that geometry was being developed and applied everywhere in the world. But this was happening in an unsystematic manner. What is interesting about these developments of geometry in the ancient world is that they were passed on from one generation to the next, either orally or through palm leaf messages, or by other ways. Also, we find that in some civilisations like Babylonia, geometry remained a very practical oriented discipline, as was the case in India and Rome.
The geometry developed by Egyptians mainly consisted of the statements of results. There were no general rules of the procedure. In fact, Babylonians and Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science.
But in civilisations like Greece, the emphasis was on the reasoning behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning. Thales, Pythagoras And Euclid. A Greek mathematician, Thales is credited with giving the first known proof. This proof was of the statement that a circle is bisected i. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent.
This process continued until BCE. Vedantu is an ardent believer of smart work and harbour experienced teaching professionals who are adept at learning and possess a greater passion for imparting the same. Vedantu makes the learning experience fun by offering solutions in a step by step explanation of numerical problems to help you improve your understanding of the concept related to the topics.
NCERT Solution of Class 9 Chapter 5 is engineered by the experts of Vedantu to serve it as an excellent material for practise and make the learning process more convenient. It is written keeping in mind the age group of the students.
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