Basic Geometrical IdeasIdeasIdeas - NCERT

1 Geometry has a long and rich history. The term Geometry is the Englishequivalent of the Greek word Geometron . Geo means Earth and metron means Measurement. According tohistorians, the Geometrical ideas shaped upin ancient times, probably due to the needin art, architecture and measurement. Theseinclude occasions when the boundaries ofcultivated lands had to be marked withoutgiving room for complaints. Construction ofmagnificent palaces, temples, lakes, damsand cities, art and architecture propped upthese ideas. Even today Geometrical ideasare reflected in all forms of art,measurements, architecture, engineering, cloth designing etc.

2 You observeand use different objects like boxes, tables, books, the tiffin box you carryto your school for lunch, the ball with which you play andso on. All such objects have different shapes. The ruler which you use, thepencil with which you write are straight. The pictures of a bangle, the onerupee coin or a ball appear , you will learn some interesting facts that will help you know moreabout the shapes around PointsBy a sharp tip of the pencil, mark a dot on the paper. Sharper the tip, thinnerwill be the dot. This almost invisible tiny dot will give you an idea of a 4 Chapter 4 Chapter 4 Chapter 4 Chapter 4 Basic GeometricalBasic GeometricalBasic GeometricalBasic GeometricalBasic GeometricalIdeasIdeasIdeasIdeasIdeas2022 -23 MATHEMATICS70A point determinesa are somemodels for a point :If you mark threepoints on a paper, youwould be required todistinguish them.

3 Forthis they are denotedby a single capital letter like A,B, points will be read as point A, point B and point course, the dots have to be invisibly With a sharp tip of the pencil, mark four points on a paper and name themby the letters A,C,P,H. Try to name these points in different ways. One suchway could be star in the sky also gives us an idea of a point. Identify at least five suchsituations in your daily A Line SegmentFold a piece of paper and unfold it. Do you seea fold ? This gives the idea of a line segment. Ithas two end points A and a thin thread.

4 Hold its two ends andstretch it without a slack. It represents a linesegment. The ends held by hands are the endpoints of the line tip of acompassThe sharpenedend of a pencilThe pointed end ofa needlelAlBlC2022-23 Basic Geometrical IDEAS71 The following are some models for a line segment :Try to find more examples for line segmentsfrom your any two points A and B on a sheetof paper. Try to connect A to B by all possibleroutes. (Fig )What is the shortest route from A to B?This shortest join of point A to B(including A and B) shown here is a linesegment.

5 It is denoted by or . The points A and B are called the endpoints of the the line segments in the figure A, the end point of each line segment?An edge ofa boxA tube lightABFig A LineImagine that the line segment from A to B ( AB) is extended beyond A inone direction and beyond B in the other directionwithout any end (see figure). You now get a modelfor a you think you can draw a complete picture of a line? No. (Why?)A line through two points A and B is written as ABs ruu. It extendsindefinitely in both directions. So it contains a countlessnumber of points.

6 (Think about this).Two points are enough to fix a line. We say two pointsdetermine a line .The adjacent diagram (Fig ) is that of a line PQ writtenas PQs ruu. Sometimes a line is denoted by a letter like l, edge of a post Intersecting LinesLook at the diagram (Fig ). Two lines l1 and l2are shown. Both the lines pass through point say l1 and l2 intersect at P. If two lines haveone common point, they are called following are some models of a pair ofintersecting lines (Fig ) :Try to find out some more models for a pair of intersecting a sheet of paper.

7 Make two folds (and crease them) to represent a pair ofintersecting lines and discuss :(a)Can two lines intersect in more than one point?(b)Can more than two lines intersect in one point? Parallel LinesLet us look at this table (Fig ). The top ABCD is flat. Are you able to seesome points and line segments?Are there intersecting line segments?Yes, AB and BC intersect at thepoint line segments intersect at A?at C? at D?Do the lines AD and CD intersect?Fig adjacement edgesof your notebookThe letter X of theEnglish alphabetCrossing-roadsFig Geometrical IDEAS73Do the lines ADs ruu and BCs ruu intersect?

8 You find that on the table s surface there are line segment which will notmeet, however far they are extended. ADs ruu and BCs ruu form one such pair. Canyou identify one more such pair of lines (which do not meet) on the top ofthe table?Lines like these which do not meet are said to be parallel; and are calledparallel , discuss and writeWhere else do you see parallel lines? Try to find ten two lines ABs ruu and CDs ruu are parallel, we write ABs ruu || CDs two lines l1 and l2 are parallel, we write l1 || l2 .Can you identify parrallel lines in the following figures?Ray of lightfrom a torchSun raysBeam of light froma light RayRail linesThe opposite edges of ruler (scale)The cross-bars of this window2022-23 MATHEMATICS74 The following are some models for a ray :A ray is a portion of a line.

9 It starts at one point (called starting point orinitial point) and goes endlessly in a at the diagram (Fig ) of ray shown here. Twopoints are shown on the ray. They are (a) A, the startingpoint (b) P, a point on the path of the denote it by APu , discuss and writeIf PQu ruu is a ray,(a)What is its startingpoint?(b)Where does the pointQ lie on the ray?(c)Can we say that Q isthe starting point of thisray?Here is a ray OAu ruu (Fig ). It starts at O and passesthrough the point A. It also passes through the point you also name it asOBu ruu? Why?OAu ruu and OBu ruu are same we write OAu ruu asAOu ruu?

10 Why or why not?Draw five rays and write appropriate names for do the arrows on each of these rays show?EXERCISE the figure to name :(a)Five points(b)A line(c)Four rays(d)Five line the line given in all possible (twelve) ways, choosing only two letters at atime from the four the rays given in thispicture (Fig ). T a starting point of eachof these rays?2022-23 Basic Geometrical the figure to name :(a)Line containing point E.(b)Line passing through A.(c)Line on which O lies(d)Two pairs of intersecting many lines can pass through (a) one given point? (b) two given points?