1. Basics of Geometry

Introduction
The term ‘Geometry’ is the English  equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’ means Measurement. Here, we will discuss some basic concepts in geometry.    

Circles

Circle                                              
A circle is a simple closed curve all of whose points are at the same distance from a given point O in the same plane . The given point O is called the centre of the circle.
It has some very special properties.

Parts of a circle
Here is a circle with centre C (Fig ) A, P, B, M are points on the circle. We will see that 
CA = CP = CB = CM. 

Each of the segments CA , CP , CB, CM is radius of the circle. The radius is a line segment that connects the centre to a point on the circle. CP and CM are radii (plural of ‘radius’) such that C, P, M are in a line. PM is known as diameter of the circle. Diameter is a double the size of a radius.
PB is a chord connecting two points on a circle.
An arc is a portion of circle. 

If P and Q are two points we get the arc PQ. We write as . 
As in the case of any simple closed curve we can think of the interior and exterior of a circle. A region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides is called a sector.

A region in the interior of a circle enclosed by a chord and an arc is called a segment of the circle. The distance around a circle is called its circumference. 

 

Quadrilaterals

QUADRILATERALS                               
A four sided polygon is a quadrilateral. It has 4 sides and 4 angles. 
We named the vertices in a cyclic manner.
This quadrilateral ABCD figure has four sides AB, BC , CD and DA. It has four angles A , B, C and D .
Quadrilateral ABCD, AB and BC are adjacent sides. 
AB and DC are opposite sides. A and C are said to be opposite angles; 
similarly, D and B are opposite angles.
Naturally A and B are adjacent angles. 

 

Triangles

TRIANGLES
A triangle is a three-sided polygon. In fact, it is the polygon with the least number of sides. 
We write DABC instead of writing Triangle ABC. The three sides of the triangle are AB , BC and CA . The three angles are BAC, BCA and ABC. The points A, B and C are called the     vertices of the triangle.

Being a polygon, a triangle has an exterior and an interior. In the figure (ii), P is in the interior of the triangle, R is in the exterior and Q on the triangle.

Angles

ANGLES                                         
 An angle is made up of two rays starting from a common end point. The two rays forming the angle are called the arms or sides of the angle. The common end point is the vertex of the angle.
This is an angle formed by rays   and  . Angle POQ is thus a better way of naming the angle. We denote this by POQ.


In specifying the angle, the vertex is always written as the middle letter.


 

Polygons

Polygons
A figure is  a polygon if it is a simple closed figure made up entirely of line segments.
For example, triangle, quadrilateral, pentagon, etc., are all examples of polygon.

If all sides of a polygon are equal and all angles are also equal , then it is called a regular polygon.

Sides, Vertices and Diagonals                                 
The line segments forming a polygon are called its sides.

  The meeting point of a pair of sides is called its vertex.

Sides  and   meet at E, so E is a vertex of the polygon ABCDE. Points B and C are its other vertices. 
 Any two sides with a common end point are called the adjacent sides of the polygon. Sides    

 

The end points of the same side of a polygon are called the adjacent vertices. Vertices E and D are adjacent, whereas vertices A and D are not adjacent vertices.
The line joining two non-adjacent vertices of a polygon is called a diagonal. Since A and C are non-adjacent vertices, so   is a diagonal.

Curves

Curves
When we draw lines on a piece of paper without  lifting the pencil and without using a scale, the shapes that we get are called curves. 
Simple Curve : A curve that does not cross itself is called a simple curve. The figures  shown below are simple curves.

Open Curves : The figures that do not begin and end at the same point are called open curves.

Closed Curves : The figures that begin and end at the same point are called closed curves. 
For example : triangle, circle, rectangle, square, etc. are all closed figures.

Position in a figure                                               
    In a closed curve, thus, there are three parts.
    (i)     interior (‘inside’) of the curve
    (ii)     boundary (‘on’) of the curve and
    (iii)     exterior (‘outside’) of the curve.
    In the figure, A is in the interior, C is in the exterior and B is on the curve.
    The interior of a curve together with its boundary is called its “region”.

Ray

RAY                                            
A ray is a portion of a line. It starts at one point (called starting point) and goes endlessly in a direction.
Look at the diagram of ray shown here. Two points are shown on the ray. A is the starting point,  P is a point on the path of the ray. 

Parallel Lines

(b)     Parallel Lines : If two or more lines do not meet each other however far they are extended, then they are called paralle lines.  


(c)     Collinear Points : Three or more points in a plane are said to be collinear if they all lie on the same line. 
In Fig, points A, B, C and D are collinear because only one line l passes through all of them.

If the points do not lie on a line, they are called non-collinear points. 

Ex     In Fig. name :                            
        (i)     Four non-collinear points.
        (ii)     Point of intersection of the lines l and m. 
        (iii)     Point of intersection of the lines r and n. 
        (iv)     Point of intersection of the lines q and n.
        (v)     Point of intersection of the lines p and q. 
        (vi)     Four line segments. 
        (vii)     Two points on the line q. 

Sol.     (i)  A, B, C, D         (ii) B        (iii) D         (iv) C         (v) A

Intersecting Lines

(a)     Intersecting Lines : If two or more lines meet each other at one point then they are called intersecting lines. Two intersecting lines have one common point.

A line

LINE                                            
 A line through two points A and B is written as   . It extends indefinitely in both directions. So it contains a countless number of points. (Think about this). Two points are enough to fix a line. We say ‘two points determine a line’. The adjacent diagram (figure) is that of a line AB written as   . 

Sometimes a line is denoted by a letter like l, m.

A Line Segment

LINE SEGMENT                                              
A line segment is part of a line. It has two endpoints and has fixed length.
We name the segment by its endpoints.
Point A and B are the two endpoints of the line segment AB as shows here.

Points

POINTS
A point determines a location. If we mark three points on a paper, we would be required to distinguish them. For this they are denoted by a single capital letter like A, B, C.
These are some models for a point :  

These points will be read as point A, point B and point C.           Of course, the dots have to be invisibly thin.
 A point has no length and no breadth.

 

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