6th RS - Fundamental Concepts PDF

. ------------ -r--- -;: ~ -------- ~ -· t>=~ fFwliildamernt al ©ealililetrriea I ©.a·m INTRODUCTION The word Geometry has been derived f...

. ------------ -r--- -;: ~ -------- ~ -· t>=~ fFwliildamernt al ©ealililetrriea I ©.a·m INTRODUCTION The word Geometry has been derived from two Greek words, namely 'Geo' and 'metron'. Here 'Geo' means earth and metron means measurement. Thus, the word geometry means, measurement ofearth. In ancient times geometry was used in land measur~ment. But, now it is widely used in various fields of our daily life. In t~s chapter, we _will learn about the fundamental concepts of geometry such as point, hne segment, hne, ray, angle, curves, triangles quadrilateral, circles, etc. ' POINT / A point is a mark ofposition. Usually, a fine dot marked with a sharp-edged pencil on a plane paper, / A represents a point. We denote a point by a capital letter A, B, P, Q, etc. In the adjoining figure, A is a point. A point has neither length nor breadth nor depth (or thickness). LINE SEGMENT / Let A and B be two points on a paper. Then, the straight path from A to B is called the line segment AB, denoted by AB. We may also call it the line segment BA, denoted by BA. A B Thus, line segment AB is the same as line segment BA. The points A and B are called the end points of AB. The distance between the points A and B is called the length of AB. Thus, a line segment has a definite length, which can be measured. How to Measure a Line Segment? For measuring a line segment, we use a ruler. One edge of a ruler is marked in centimetres (cm) and millimetres (mm). Example I. Measure the length of the line segment PQ, given below: P- - - - - - - - - - - Q Method : Place the edge of the ruler along the given line segment PQ, keeping the f 0 cm mark of the ruler at the point P, as shown below. ""I"';,' '"'";;I"111'"~1''' '"7'"I' '~I11 ~··~1"' 'JI ~I'' "I' '~I''''I'''~'''' I';·~ \ Read the mark on the ruler against the point Q. We find that the point Q is 2 small divisions ahead of the 5 cm mark. :. Length of PQ = 5.2 cm. ~ l~ / I,.. le tJ rrie se gmen t extended endlessly on both sides is called a llne. / ft ~ 9egxnent ~ e)cte ed on.22th s~es and marked by arrow mark s at nd the two ~ u.iie nts a lme, denot ed by AB or BA. epres e ◄ ,►.,.&s t h 5 no end point s. A e e line a ft 't d" · / af!:OW ~ in Qppgsi -~ i~~ctions of a line indicate that it exten ds indef initel y in 'f\110 ~io ns. - i,otb_ t~ [ine does ~ot have a ~fini te length. repre sent / ··. ce the line is endless, it canno t be drawn on a paper. We can simp ly Sill hown above. JjJlB ass. 11 draw ing a lme AB would mean to _draw a ~art of it, mark two point s A and Bon Br put arrow mark s at the two ends m opposite directions. ◄ , , ►I it 811 t'9'1"1es we may denot e a line by a small letter A e SoIIle l.u-- ' l, ,n, n, etc... In the given figur e, l is a line. RA! line segment extended endlessly in one direction is called a my. A }ine segm ent AB exten ded in the direction from A to B B and mark ~ by an arrow mark at B, repre sents a ray AB, A denoted by AB· A ray AB has one end point , namely A. This end point A of the ray AB is called its initial point. can simp ly Since a ray is endle ss in one direction, it cannot be drawn on a paper. We represent a ray as show n above. Clearly, a ray has no definite length. AB is a ray with initia l point A and extending endlessly in the A B direction from A to B. It is repre sente d by the adjacent figure. BA is a ray with initia l point Band extending endlessly A B in the direc tion from B to A. It is repre sente d by the adjacent figure. Clearly, ray AB and ray BA are two different rays. Oppo site Rays ite direc tions Two rays with the same initia l point and extending indefinitely in oppos along the same line, are called opposite rays. ,,. In the adjoi ning figure , QA and OB are two rays ◄ B I I V 0 A with the same initia l point O and exten ding infini tely in opposite direc tions along the same line l. So, QA --- are oppos ite rays. and OB ( 1-). , '1 I r, 1I j I I ' I I { 1) f' r I I I Distin ction Betwe en a Line Segm ent a Line and a Ray ' Line segm ent Line Ray (i) A line segme nt has two A line has no end point. A ray has one end point. end points. (ii) A line segme nt has a A line does not have a A ray does not have a defini te length. defini te length. defini te length... (iii) A line segm ent of a A line canno t be drawn on A ray canno t be drawn tven lengt h can be a paper. We can simpl y on a paper. awn on a paper. repres ent a line. ' We can simply represent a ray. (iv) A B is a line A line AB is repres ented A ray AB is represented segme nt. by ◄ A ► by A ► B B (v) Line segme nt AB is the Line AB is the same as same as line segme nt line BA. ~ s AB and BA are erent. BA. SURF ACE A solid has a surface which may be curved or fiat. For examp le, the surface of the ball is curved while the surface of the wall is flat. PLAN E / A plane is a flat surface which extends indefinitely in all the directions. The surfac e of a smooth wall, the surface of the top of the table, the surface of a smoo th blackb oard, the surfac e of a sheet of paper, the surface of calm water in a pond are all examp les of a portion of a plane. Since the plane is endless in all directions, it canno t be drawn on a paper. In practi ce, only a portion of a plane is drawn and it is repres ented by a rectangle or a parall elogra m. A plane is named by taking three different points on it, which are not on the same line. In the given figure, ABC is a plane. Intersecting Lines / If there is a point P common to two lines l and m in the same plane, we say that the two lines intersect at the point P and this point P is called the point of intersection of the lines. lparatle l Lines / {; Two lines l and m in a plane are called paralle l lines if thei do not intersec t even when produce d and e write, l II m. The distanc e betwee n two parallel lines ~ m ~ways remain s constrn t. Opp~ edges of a ruleiJ opposit e edges of the top of a table, Railway tracks, etc., are exampl es of paralle l lines;_J_, @oncu rrent Lines.,,,/ ' Three or more lines in a plane are said to be concurr ent if all of them pass through the same point. ~-~~-♦,, This commo n point is called the point of concurr ence of the P given lines. In the adjoini ng figure, the lines l, m, n, pall pass through the point P. So, all these lines are concur rent and P is the point of concurrence of these linesJ @olline ar Points / Three or more points in a plane are said to be colline ar if all of them lie on the same straigh t line. ► I I I I I I ► A B C D p Q R S (i) (ii) In fig (i), all the four points A, B, C, Dare collinea r. In fig (ii), the points P, Q, R, Sare non-coll inear. J.CURVES The figures traced out with the help ofthe sharp edge ofa pencil without lifting the pencil_are called curves. Each of the figures shown below is a curve. (i) (i i) (iii) (iv) (v) (ui) Closed and Open Figure s A figure which begins and ends at the same point is called a closed figure. In the above figure, curves (i), (ii), (iv) and (vi) are closed figures. A figure which does not end at the starting point is called an open figure. In the above figure, curves (iii) and (v) are open figures. POLYGONS A simple closed figure formed of three or more line segments is called a polygon. The line segmen ts which form a polygon are called its sides. The point at which two adjacent sides of a polygon meet is called vertex of the polygon. I A polygon with 3 sides is called a triangle. I A polygon with 4 sides is called a quadrilatera l. A polygon with 5 sides is called a pentagon. A polygon with 6 sides is called a hexagon; and so on. Triangle Quadrilateral Pentagon Hexagon PLANE FIGURES We draw figures such as a triangle, a rectangle, a square, and a circle, etc. We call these figures as plane figures. Triangle : A figure bounded by three line segments is called a triangle. In the adjoining figure, ABC is a triangle, written as MBC. A This triangle has : (i) Three vertices namely, A, B and C; (ii) Three sides namely, AB, BC and CA. B C Quadrilat eral : A figure formed by four line segments is o□: called a quadrilate ral. In the given figure, ABCD is a quadrilatera l. It has four sides namely, AB, BC, CD and DA. A It has four vertices namely, A, B, C and D; Various Types of Quadrilaterals D. - - - - - - -.C Rectangle : A quadrilateral bounded by two horizontal line segments AB and DC and two vertical line segments BC and AD, is called a rectangle ABCD. A,____ _ _ _...JB In the given rectangle ABCD, we have (i) AB =CD and BC =AD. (ii) LA= LB= LC= LD = 90°. Note : (i) The opposite sides ofa rectangle are equal. (ii) Each angle of a rectangle measures 90°. Square : A quadrilateral hav~ng all sides equal and all angles equal, is called a square. In the given figure, PQRS is a square in which PQ = QR= RS= SP and LP= LQ =LR= LS= 90°. Parallelogram :A quadrilateral in which the opposite sides DP a are parallel, is called a parallelogram. ~  In the given figure, ABCD is a quadrilateral in which AB 11ocsndAD II BC. In a parallelogram : (i) the opposite sides are parallel and equal..·.AB= DC and AD= BC. (ii) the opposite angles are equal. LA= LC and LB= LD. s Rhombus : A parallelogram having all sides equal, is called a rhombus. ------,R In the given figure, PQ 11 SR, PS 11 QR and PQ = QR= RS= PS. :. PQRS is a rhombus. CIRCLE / Activity : Place a one-rupee coin on a paper and with a sharp- edged pencil, starting from a point, trace its boundary on the paper, till your pencil reaches the starting point. You obtain a figure as / shown herewith. / ' t,1 / ' \ \\' This figure is known as a circle. I I I / I i (,,., I. I ' ).1 ' ~ \' '\;, · (~ I I ' ·' A bangle, a wheel, a coin, etc., are circular objects. INTERIOR AND EXTERIOR OF A CLOSED FIGURE The part ofthe plane enclosed by the boundary ofa closed figure is called its interior. The part ofthe plane lying outside the boundary ofa closed figure is called its exterior. Examples : Look at the figures given below : D F.-..r-..G S R C A E B (i) (ii) (i) In fig (i), the points A, B and C lie in the interior of the given figure; D and E are the points lying on the figure while F and G are the points lying in the exterior of the figure. (ii) In fig (ii), the points P and Q lie in the interior of the figure, the points R lies on the figure and the point S lies in the exterior of the figure. Linear Boundaries of a Figure If the boundaries of a figure are straight lines only, then these are known as the linear boundaries of the figure.  Exampl es: Each ofthe figures - a triangle, a rectangle, a square and a parallelogram consists of linear boundar ies. DD Figures having linear boundaries Curvili near Bounda ries of a Figure Ifthe boundar ies ofa figure are not straight lines, then these are known as curvilinear boundar ies. Examp les : Each of a circle and an oval has curviline ar boundaries. 0 A circle An oval - - - - - - - -. EXERCISE 14A - - - - - - - - 1. Identify and name the line segment s and rays in the following figures : \ (i) A C (ii) G- - - - - o , - - ---, M\ H- - - i E 0 _ _ ____, F D 2. What do you mean by collinear points? (i) Mark three points which are collinear. (ii) Mark three points which are non-collinear. Draw lines through these points taking two at a time and name these lines. How many such different lines can be drawn? 3. (i) How many lines can be drawn passing through three collinear points? (ii) Given three collinear points A, B, C. How many line segment s do they determine? Name them. 4. What do you mean by concurrent lines? Draw four concurrent lines. 5. , Count the number ofline segments drawn in each ofthe following figures and name 1 them: (i) (ii) (iii) p Q R s \ 8. ,Ul in the blanks : (s) A line segmen t has __._1.... 1,,"""'''-- - - end points. (ii} A line has fr, /-z -c C, 1 end points. (iii) A ray has tw, (! end point. (iv) A line segmen t has a dof; t1 dr length. (v) A line AB is represe nted by _ tf (vi) A ray AB is represe nted by -.iir, ~...._ ___ _...._' - - - - - - (viiJ A point shows a definite ,-, 0 , ,I, pq I (viiiJ Two lines interse ct in a_ f' ,~.J "f ,.,h 1c,(,OJ,. (~) Two planes interse ct in a _ 1 -'--;=n_,_f_ _ __ (x) The minimu m numbe r of points of intersec tion of three lines in a plane is "2~.,~ lo. (xi) The maxim um numbe r of points of intersec tion of three lines in a plane is 3/ tl,,ye e 7, iook at the adjoini ng figure carefull y and name: (i) all pairs of paralle l lines; (ii) all pairs of interse cting lines and their points of intersec tion; (iii) all sets of colline ar points; (iv) concurr ent lines and their points of concurr ence; p (v) one set of three non-collinear points. 8, Which shape do the followin g have? (i) The wheel of a bicycle (, ) l If (ii) Ablackboard R, , 1u , JI e (iii)A page of a book ~ ~ 0 1'' ··} ~ (iv) The face of the full moon ( , r c /e (v) A set-squ are in a geomet ry box (vi) A face of a dice Set l, ci t C ~ i () ~) c...l ( 9, Fill in the blanks : J (i) A triangle has 'Z sides and _ _ J._____ vertices. (ii) A rectang le has Lf sides and Lf vertices. (iii) All the sides of a square are Q,,