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Basic Geometrical Ideas

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Published in: Mathematics
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This Note will help class 6th students to get basic understanding of different geometrical shapes.

Amit A / Ahmedabad

7 years of teaching experience

Qualification: B.Sc (Magadh University - 2008)

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  1. CHAPTER-4 BASIC GEOMETRICAL IDEAS Refe re n c e NCERT
  2. Introduction of geometry Geometry is used to measure shapes and size of different objects such as buildings, cupboard, fruits box, shoes box etc., Geometry comes from Greek word Geometron as Geo- Earth and metron- Measurement. Points Point represent the basic location of any structure any shape starts from a point and end to a point. A point basically determines a location Points are represented by a thin and small tiny dot. Which you can make with tip of your pen or pencil on a piece of paper as shown here. connecting different points through line different shapes are formed. > Therefore it can be said that point is the origination of any structure in universe.
  3. Points are denoted by English alphabets A to Z B both in upper case and lower case. Distance between two points A and B makes a Line Segment as shown here and it is B represented as AB point A and B are called end point of line segment. Line Segment : Line segment between point A and B is written as AB but it can be extended indefinitely in both the directions. So a line can contains countless number of points. Two points are enough to form an Line. Two points together determines a line p The diagram represents a line PQ and can be written as PQ line can also be denoted by letter like l, m etc.
  4. You may find various examples of line segment surrounding you such as Tube light, edge of a rectangular table, edge of your text book etc. Line Segment Tube Light Line Segment Top View of Table Book You can see that by connecting various dots with line segments and connecting line segments various shapes can be formed. As you can see here in the figures of different objects.
  5. Intersecting lines The point at which two or more different lines meets or cross each other is called intersecting point of the lines and this is called Intersecting Lines. For example look at the diagram below. p 6 n You may also find examples of intersecting lines in your surroundings. For example . Two line segment hand b are intersecting each other at point P. if two different lines have a common point then they are called intersecting lines. B O m Here you can see a rectangular table top which is made by joining 4 different Line segments named l, m, n and o
  6. B You can see that line segment I and n are intersecting at point A line segment I and o are intersecting at point B line segments m n m O and n are intersecting at point c and line segment m and o are intersecting at point D D. Here you can see that 4 different lines segments l, m, n and o are intersection in such a way that that they are forming a rectangular shape which is top of a table. 'While going to your school you may have passed through a cross road. Means a location from where 4 different roads connect through. one of which may got to your school. 'You may have also seen a circular boundary at the centre also called as "Golumber" in local language. This "Golumber" can be considered as the intersecting point of the roads. 'Look at the figure adjascent representing cross road. "Golumber"
  7. Parallel Lines When two or more different straight lines are drawn opposite each other in a horizontal or in a vertical manner in such a way that they do not intersect then they are called Parallel Lines When you observe different objects in your house or school you will find various examples representing Parallel Lines. For example you may have seen bamboo ladder in you house lets understand it in adjacent figure of ladder. B c Here you can see that line segment AB and CD are parallel to each other and it is represented as AB Il CD a b c m Observe this ladder you can see that side I and m are parallel to each other and line a to e are also parallel to each other. Where as line segment a to e are intersecting at various points on side I and m of the ladder.
  8. Some more examples of parallel lines around you B You may have seen a rectangular desk kept in front of your class room which near your teacher stand. Now observe this table in the adjacent figure. c When you observe this table you can see that the flat top of this table is made by joining points ABCD with straight lines and these lines are intersecting each other at points A, B, C and D respectively. Now you can see here that line AB and BC are intersecting at point B, line AB and AD are intersecting at point A, line DC and AD are intersecting at point D and line DC and CB are intersecting at point C. Now when you observe the 4 legs of the table you will see that leg AH is intersecting at point A, leg BG is intersecting at point B, leg DE is intersecting at point D and leg FC is intersecting at point C. So you can see here that in case of this table points A,B,C and D are intersecting point of 3 different lines.
  9. B When you observe this table you will also find that line side AB parallel to side DC and side AD is parallel to side BC and this can be represented as AB DC and AD BC c We can also see here that all the 4 sides representing legs of the table are also parallel to each other and can be represented as AH BG I DE I CF Similarly may also find various examples of parallel lines like railway track, Scale in your pencil box, Opposite sides of your bed etc. Rays: Rays are lines that originate from a source of light which starts from a point and has no end. Means it has no end point it goes to infinite. Rays can be represented as below. p Here P denote the starting point of the ray from where it originates i.e., the source of the light and the arrow at the top of this line segment represents that it will go to infinity, means it has no end point. Examples from where rays may originate are Torch, Bulb, Tube Light, and SUN which is the natural source of Rays.
  10. In previous discussion we have seen that various points can be taken on a line segment so we can also take various points on a ray. As shown here in the figure Points O and m. p In case of representing Rays here we can represent this rays as PO and PM both are representing the same ray which is originating from point P but we cannot represent this ray as 0M as point O is not the source of the Ray. ' In case of representing rays originating point of the ray has to be taken into the consideration. We can represent this ray as PO or PM but we cannot represent it as 0M, MO, MP, OP because this denotation is not starting from Origination of the Ray which is point P. 'There are countless numbers or rays arises from it originating point for example you see that SUN is the natural source of light and it originates rays which reaches to different planet in the universe including earth on which we leave.
  11. p c c B B In the figure there is an originating point of Rays named as P. Name the rays originating from point P. Solution: You can see there that there are two different ray originates from point P and can be represented as PA PB respectively where P and PB represents the same ray having two different points C and B Answer these for adjacent figure: (a) Name fine points Ans- A, B, C, D, E (b) A line segment Ans- AD ( Because it is represented by arrow on both the side and can be extended as long as you want) (C) Name the 4 rays in figure Ans- Look at Point C there are 4 rays appearing from point C and there name are — CD, CE, CA and the ray going upward from point C
  12. ' There are unlimited lines that can be drawn through a given point P. ' Only one line can be drawn through two given points. Draw Line PQ intersecting line AB at point M o p p
  13. Curves Two or more points not connected through a straight lines are called as a curved line as shown in adjacent figure. Here you can see the way points A and B are connected this pattern is called as a curve. The are three different types of curves Simple Curve, Open Curve and B Closed Curve. Simple Curve If curve does not cross itself at any point than it is called a Simple curve. Example as in below figure
  14. Open Curve Closed Curve o c Curve which has two different points not connected with each other as shown below. B Curve which has no open point is called a closed curve as shown below. A closed curve have three different parts represented by points in below figure Point A is on the curve. Point B is inside the curve and point C is outside the curve. The interior of the curve together with its boundary is called "region"
  15. Polygons Shapes that are formed by joining 3 and more than 3 straight lines together at different points are called polygons. Polygons are two dimensional in nature. Examples of polygons as below A zoo (c) (d) These shapes shown above are polygons. Figure (a) above is the simplest form of polygon and also called triangle. So we can also say that polygons are simple closed figure made of joining line segments together at different points.
  16. 'Line segments which forms a polygon are called "sides" of polygon. 'The point at which a pairs of line segments meets is called "Vertex" and all the points all together of a polygon are called as "vertices" of the polygon. 'When two non adjacent points of a polygon is joined through a straight line this straight line is known as "Diagonal". 'Lets try to figure out these parts of a polygon in below diagram. Side Diagonal B c Vertex 'AB, BC, CD and DE are sides 'Point A, B, C, D and E altogether are called Vertices AD, AC and EB are diagonals
  17. Angles Points at which two or more lines meet, angles get formed over their. As shown in this figure. p Ve rtex Here you can see that line OP and ON are meeting at point O and forming an angle at point O this angle is represented as LPON. The point at which angle is formed is always kept in the middle. Angle between two lines is denoted by a curved line as shown in the figure. Point O is also called vertex of the angle.
  18. How many angles can you identify? Exterior p B Interior c Here you can see that 3 different rays AP, BP and CP are meeting at point P so this is the point where angle is formed. There are 3 angles formed in this case and will be named as LAPB, LCPB and LAPC When two or more lines meet at a point two different angles are formed and are called as Interior and Exterior angle. The angles has three different part associated with it Interior. Exterior and on the angle itself.
  19. Angles are also formed in closed figures like rectangle, squares, triangles etc as shown below. B c Fig-I Q p Fig-2 In figure 1 you can see that there is a rectangular closed figure with 4 sides and these sides forming 4 angles at point A, B, C and D respectively. Angles can be represented as LA, LB, LC, and LD . These angles can also be represented as LCAD, LABD, LBDC and LACD In figure 2 you can see that there is a triangular closed figure with 3 sides and these sides forming 3 angles at point P, Q, and R respectively. Angles can be represented as LB LQ, and LR or LQPR, LPQR and LPRQ
  20. Triangles Triangle is three sided polygon. It is a polygon with lowest number of sides. Here is a triangle made with 3 different lines AB, AC, and BC intersecting at points A, B and C respectively. This triangle can be represented as AABC A, B and C are called Vertices of the triangle where angles are formed as LBAC, LABC, and LACB Because it is a polygon it has interior and B exterior part. In figure you can see that point P is in interior part of triangle where as point Q is at exterior part of the triangle. p c
  21. B How many triangles are their in below figure? There are 3 triangles altogether in this figure : AABC, AABD and AADC c How many angles are there in this triangle? There are total 7 angles in this triangles named as LABD, LBAD, LADB, LDAC, LADC, LACD and LBAC
  22. Quadrilaterals Quard = 4 and Lateral = Sides Quadrilaterals = A polygon having four sides and 4 angles B Here is a quadrilateral having 4 sides AB, BC, CD, and DA and 4 angles LA, LB, LC and LD respectively About Quadrilaterals 1. 2. c 3. 4. Quadrilaterals are always named in a cyclical manner as ABCD. AB, DC and AD, BC are the opposite side of the quadrilateral LA, LC, and LD, LB are opposite angles of quadrilateral. LA, LB and LD, LC are adjacent angles of quadrilateral.
  23. Circles You may have seen various circular objects around you which represents examples of circle. For example Bangles, Wheels of your Car or Motor Bike etc., B p Parts of a Circle c 'Here is a circle with points surrounding it named as A,B,C,D,E,F,G and H respectively. Point P is the centre of the circle. are called radius of the circle. Means a radius is line segment that connects the centre of the circle with a point E on the circle. All the radius together is called as radii (Plural of radius). ' Radius of circle is denoted by letter r 'Line segment AE, BF, CG, DH, EA, FB, GC, HD are Diameters of the circle. Means a straight line which passes through the center of the circle and connect the two opposite points on the circle is called Diameter.
  24. Arc B p Fig-I x Fig-2 c D E Radius Chord 'Diameter of circle is denoted by letter d. ' Diameter is double the side of radius. Means from figure PB + PF = Diameter. r + r = d also written as 2r = d Line segment AB, BC, CD, DE, EF, FG, GH and vice versa are Chord of the circles. A line segment that connect two different points on the surface of the circle and this line should not passes through the center of the circle is called Chord. 'Diameter of a circle is also a chord and it is the longest chord of a circle.l ' The curved part between any two point of the circle is called Arc. Such as the curved part between point A and B or Point F and H is called arc and is denoted as AB or FH ' In fig-2 the shaded area lying between the two radii PQ and PR and Arc RQ is called Sector of the circle. 'In fig-2 the shaded area lying between chord XY and Arc formed between XY is called segment of the circle.
  25. p B 'Measurement of length from point A to Point A through B,C and D is called Circumference of the circle. Means the distance around a circle is called Circumference of the circle. As explained in fig-I p C o Fig-2 Fig-I Semi-Circle ' Diameter of a circle divides it into two equal halves. Each half part is called Semi-Circle. As shown in fig-2 Diameter
  26. Part of Circle at a glance Arc Semi-Circle Segment c Diameter Center Sector Chord 'Two diameter of a circle will always intersect each other at the center of the circle. 'A diameter of a circle divide the circle into two equal halves which forms semi-circle.
  27. Thank You

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