Introduction to Euclid’s Geometry Class 9 Notes Maths Chapter 5

Students can go through AP 9th Class Maths Notes Chapter 5 Introduction to Euclid’s Geometry to understand and remember the concepts easily.

Class 9 Maths Chapter 5 Notes Introduction to Euclid’s Geometry

• ‘Geo’ means ‘earth’ and ‘metron’ means ‘to measure’.
• ‘Geometry’ is an important ancient branch of mathematics.
• Geometry deals with
  1. Area of given
  2. Volumes of solid figures
  3. Approach of required constructions
  4. Planning and extracting the constructions
  5. Estimates / calculates the length of sides, measurement of different angles etc.
  6. Shape of individual objects
  7. Properties of different shaped objects (Polygons etc.)
  8. Spacial relationship with objects.
• Mensuration is a sub-branch of Geometry.
• ‘Sulba Sutras’ were the manuals of geometrical constructions in our ancient India.
• In the Vedic Period also, our ancestral Indians had the knowledge of Geometry and established some rules for construction of “Altars”.
• ‘Altar’ is a sacred space.

→ In Vedic Period :

• For household rituals, square and circular shaped ‘altars’ were used.
• For ‘public’ rituals, combination of rectangles, triangles and trapezium were in practice.
• ‘SRIYANTHRA’ is a complex geometrical pattern that is said to have the power to manifest abundance and happiness in the life of every human.
  Particularly this “Sriyanthra” is used to symbolize or represent the universe and the divine feminine energy.
• ‘SRIYANTHRA’ consists of ‘9’ (Nine) interwoven isosceles triangles that are arranged in a way that these 9 triangles produce 43 subsidiary triangles.
• It is believed that this “Sriyanthra” is a powerful tool to bring success, health and wealth in one’s life.
• In sacred geometry the interwoven triangles can be seen as a symbol of balance, harmony and the inter-, connectedness of all things. Also the 3 sides of these triangles represent 1) Spirit, 2) Mind, 3) Matter.
• Interwoven triangles represent the union of ‘male’ and ‘female’ eriergies.

Mind mapping
→ Background :

• It is named after great Greek Mathematician “EUCLID”.
• One of the ancient branches of Vedic mathematics.
• Deals with the points, lines, angles, shapes, 2D, 3D spaces.

→ Key terms and their definitions :

1. Points: A location in space with no size or dimension.
2. line : A continuous series of points in space.
3. Angle: The space between two lines at common end point.
4. Shape : An arrangement of points (or) lines in specific form.

→ Axioms and postulates:

1. Axiom : Axiom is a mathematical statement that is assumed to be true without proof.
   It is considered to be self evident.
2. Postulates : Postulate is a geometrical statement that i^ assumed to be true without proof.

→ Thales:
Thales is a Greek mathematician.
He gave first (known) proof.
He proved that a diameter bisects the circle.

→ Pythagoras:
Pythagoras.was student of Thales.
Pythagoras developed the theory of geometry to a great extent.
He is familiar with Pythagoras theorem. Foot prints of Pythagoras in geometry are very important.

→ Euclid:

• Basically “Euclid” was a teacher of maths, who had a great interest in maths.
• He collected all possible information about mathematics particularly in geometry and put it all in an organised manner.
• He wrote “Elements” named book.
• “Elements” book is treated as “Holy Book of Geometry”.

→ Basic (Key) words :

• Space : A space is that which do not have shape, size, position.
• Solid : A solid is that which have some shape, size, position and can be moved from one place to another place.
• Surface: Boundaries of solids are called surfaces.
• Surfaces separate spaces.
• We consider, the surfaces won’t have any thicknesses.
• It has length and breadth only but not have any thickness.
• The edges of surfaces are lines.

→ Plane surface : A plane surface is a surface which lies evenly with straight lines on itself.

→ Point : A point is that which has no part. It means we cannot cut it into sub parts.
Line : A line which is formed by joining points. It has no breadth.

→ Straight line : It is a geometrical shape that extends infinitely in both directions and has no curvature.

• A straight line by which lies evenly with the points on itself.

• Actually we consider a point; a line and . a plane as undefined terms. We can explain them with the help of “Physical models”.

→ Some important Euclid’s axioms :
1. It equals are added to equals, then the wholes are equal.
Ex : If a = b and c = d, then a + c = b + d.

2. Things which are equal to the same thing are equal to one another,
Ex : If a = c and b = c, then a = b.

3. If equals are subtracted from equals the remainders are equal.
Ex: If b = c, then b – a = c – a.

4. Things which coincide with one another are equal to one another.
It means that if two objects occupy the same position or space, then they must have same shape and size.
Ex:

Two rectangles of same measure-ments or two squares of same sides etc.

5. The whole is greater than its part.

6. Things which are double of the same things are equal to one another.

7. Things which are halves (Triples) , (quarters) of the same things are equal to one another.

→ Euclids postulates:
1. A straight line may be drawn from any point to any other point.
From this postulate, we can conclude following axiom.
For any given two points, there is a unique line that passes through them.
Ex: P, Q are two given points, then \(\overrightarrow{\mathrm{PQ}}\) is only one straight line passing through given ‘P’ and ’Q’.

Other lines l, m are passing through ‘P’ only, whereas lines (k, s) are passing through ‘Q’ only.

2. Postulate – 2 : A terminated line can be produced indefinitely.
Explanation : Terminated line means a line segment. So, we can re-write above postulate as follows.
“A line segment can be extended both sides indefinitely”. Let \(\overrightarrow{\mathrm{PQ}}\) is aline segment as shown.

3. Postulate – 3: A circle can be drawn with any centre and any radius.
Explanation : You can consider any point to draw desired circle. That point will be centre of it.
You can consider any radius to draw a circle.

4. Postulate – 4 : All right angles are equal to one another.

5. Postulate – 5: If a straight line (crossing) falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles.
The two straight lines, if extended indefinitely meet on that side, oh which the angles are less than two right angles.
This postulate is also familiar as parallel postulate.

• This postulate essentially deals with the geometry of parallel lines. .
• It assests that if a line crosses two other lines and the interior angles on one side of the crossing (transverse) are less than 180°, then the two lines will eventually intersect on that side.
• This postulate is often contrasted with the remaining 4 postulates which are more easier and intuitive (for clear understanding). Historically it is less selfevident.

l, m, n are 3 lines, ‘n’ is crossing other two lines (l, m).
Here 1, 2 are interior angles on the same side of ‘n’.
Then if ∠1 + ∠2 < 180°, then l, m meets on the same side of ∠1, ∠2 (on extension).
So, l, m will eventually intersect on
the same side of these interior angles ∠1, ∠2.

• A system of axioms is called consistent.
• Theorem / proposition : A statement that was proved is called theorem.
• Basing on his axioms (consistent) and postulates, Euclid deduced 465 theorems.

→ To locate the exact position of a point on a number line we need only a single reference.

→ To describe the exact position of a point on a Cartesian plane we need two references.

→ Rene Descartes a French mathematician developed the new branch of mathematics called Co-ordinate Geometry.

→ The two perpendicular lines taken in any direction are referred to as co-ordinate axes. © The horizontal line is called X – axis.

→ The vertical line is called Y – axis.

→ The meeting point of the axes is called the origin.

→ The distance of a point from Y – axis is called the x co-ordinate or abscissa.

→ The distance of a point from X – axis is called the y co-ordinate or ordinate.

→ The co-ordinates of origin are (0, 0).

→ The co-ordinate plane is divided into four quadrants namely Q₁, Q₂, Q₃, Q₄ i.e., first, second, third and fourth quadrants respectively.

→ The signs of co-ordinates of a point are as follows.
Q₁: (+, +) Q₂: (-, +) Q₃: (-, -) Q₄: (+, -).

→ The x co-ordinate of a point on Y – axis is zero.

→ The y co-ordinate of a point on X – axis is zero.

→ Equation of X – axis is y = 0

→ Equation of Y – axis is x = 0

→ In a co-ordinate plane (x₁; y₁) ≠ (x₂, y₂) unless x₁ = x₂ and y₁ = y₂.