Geometry

Geometry is derived from two Latin words, geo + metron meaning earth & measurement. Thus it is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.

Why we Study about Geometry?

To find Length, Area, Volume of different Plane and Solid figures of use which are present around us in this world and to know better about them.

Geometry

Contemporary geometry has many subfields which are as follows-

(i) Algebraic Geometry – is a branch of geometry studying zeros of multivariate polynomial. It include linear and polynomial algebraic equation used for solving about the sets of zeros.The application of this type include Cryptography, string theory etc.

(ii) Discrete Geometry – is concerned with relative position of simple geometric object, such as points, lines , triangles, circles etc.

(iii) Differential Geometry – uses techniques of algebra and calculus for problem solving. The various problem include general relativity in physics etc.

(iv) Euclidean Geometry – The study of plane and solid figures on the basis of axioms and theorems including points, lines, planes, angles, congruence, similarity, solid figures. It has a wide range of application in Computer Science, Modern Mathematics problem solving, Crystallography etc.

(v) Convex Geometry – includes convex shapes in Euclidean space using techniques of real analysis. It has application in optimization and functional analysis in number theory.

(vi) Topology – is concerned with properties of space under continuous mapping. Its application includes consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.

Another way of classification of Geometry include Plane and Solid Geometry.

Plane Geometry – This wing of Geometry deals with flat shapes which can be drawn on piece of paper. These includes lines, circles & triangles of two dimensions.

Soid Geometry – It deals with 3-dimensional objects like cubes, prisms, cylinders & spheres.

Point –

A precise location or place on a plane. Usually represented by a dot. A point is an exact position or location on a plane surface. It is important to understand that a point is not a thing, but a place. It is important to note that a point has no dimension rather it has only position.

Line – is straight (no curves), having no thickness and extends in both directions without end (infinitely). It is important to note that it is the combination of infinite points together to form a line.

  • Line Segment – If a line has a starting and an end point then it is called as Line Segment.
  • Ray – If a line has a starting point and has no end point is called as Ray.

Eg. Sun Rays

Angles – In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

Types of Angle –

Acute Angle – An Acute angle meaning Sharp is an angle smaller than a right angle ie. it can range between 0 – 90 degrees.

Obtuse Angle – An Obtuse angle is more than 90 degrees but is less that 180 degrees.

Right Angle – An angle of 90 degree.

Straight Angle – An angle of 180 degree is a straight angle. Such as angle formed by a straight line

Polygon

A plane figure that is bounded by a finite chain of straight line segment closing in a loop to form a closed polygonal chain or circuit.

The name ‘poly’ refers to multiple. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon polygon.

General Formula for Sum of internal Angles of a polygon –

Sum of internal Angles of a polygon = \((n-2)\times 180\)<

Different types of Polygon –

Polygon type Definition & Property Types
(i) Triangle – A 3-sided polygon whose sum of internal angle always sums to 180 degrees.
  • Equilateral Triangle – Has 3 equal sides and angles.
  • Isosceles triangle –  Has 2 equal sides and angles.
  • Scalene triangle – Has all the 3 unequal sides and angles.
(ii) Quadrilateral A 4-sided polygon with four edges and four vertices.

Sum of internal angles is 360 degrees

  • Square – Has 4 equal sides which are at right angles.
  • Rectangle – Has equal opposite sides and all angles are at right angles.
    • Parallelogram – Has two pairs of parallel sides. The opposite sides & opposite angles are equal in measure.
  • Rhombus – Has all the four sides to be of equal length. However they do not have its internal angle to be 90 degrees
  • Trapezium – Has one pair of opposite sides to be equal.
(iii) Pentagon A plane figure with five straight sides and five angles
(iv) Hexagon A plane figure with six straight sides and six angles
(v) Heptagon A plane figure with seven sides and seven angles
(vi) Octagon A plane figure with eight straight sides and eight angles.
(vii) Nonagon A plane figure with nine straight sides and nine angles.
(viii) Decagon A plane figure with ten straight sides and angles.

Circle –

A Circle is a simple closed shape. From a certain point called centre, all points of a circle are of same consistent distance ie. the curve traced out by a point that moves so that its distance from centre is constant .

Circles

Understanding Similarity and Congruence –

Similarity – Two figures are said to be Similar if they have same shape or have equal angle but do not have same size.

Congruence –  Two figures are said to be Congruent if they have same shape and size. Thus they are totally equal.

 


Basics of Geometry:

The space which surrounds us consists of many objects. The branch of mathematics dealing with shape, size, and relative position of these objects is known as Geometry. It is derived from the Ancient Greek word ‘Geometron’ where ‘Geo’ means ‘Earth’ and ‘Meteron’ means ‘Measurement’.

Point: A point is a zero dimensional figure used to specify the position or location. It has no length or breadth and is represented by a dot as shown in the figure, P represents a point.

Geometry

Line: A one-dimensional collection of points which extends infinitely in both the directions is a line. It has no thickness and it is straight.

The figure below represent a line and the two arrows indicate that it extends infinitely.

Geometry

The line shown above is represented as\(\overleftrightarrow{PQ}\) or m.

Line Segment: Line segment is any portion of a line which has two end points.

The shortest distance between two points, is measured using a line segment. If we have two points A and B, as shown then the shortest distance AB between them is measured with help of line segment \(\overline{AB}\).

On extending any line segment from both the ends infinitely, we get a line.

Geometry

A line segment always has two end points. The following models will help you understand better:

geometry

Geometry

Curve: When a point moves in a random manner such that it does not follow a straight line path then the path traced by the point forms a curve.

Geometry

Intersecting Lines: Two lines which cross each other at a point are said to be intersecting each other at that point. Consider two lines l and m, meeting each other at point O.

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Two lines intersect each other if they have one common point between them. Hands of a clock at different positions are an example of intersecting lines.

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Parallel Lines: Two lines that never intersect each other, irrespective of how far they are extended are known as parallel lines.

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Two parallel lines  a and b are represented as a||b . Railway tracks, opposite edges of a ruler are examples of parallel lines.

Geometry

Ray: A portion of line with one distinct end point is known as a ray. The figure below shows a ray, represented by \(\overrightarrow{OP}\).

Geometry

Light rays coming from sun, torch, and light house are examples of ray.

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Angle: If two rays have a common end point, then they form angle between them. In other words, if a ray is rotated in such a way that it completely coincides with the direction of other ray, having the same end point, then the measure of rotation of the ray gives the angle between them.

Geometry

The angle formed by rays \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) is given by \(\angle{AOB}\) or \(\angle{BOA}\). \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\)  form the arms or sides of the angle and  is the vertex.

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Practise This Question

An exact location in space is called a _______