1 2 GEOMETRY GEO METRON GEO THE EARTH

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GEOMETRY = GEO + METRON GEO- THE EARTH METRON- MEASUREMENT GEOMETRICAL IDEAS ORIGINATED DUE TO THE NEED FOR MEASUREMENT, ART, CONSTRUCTION. 1. FOR MARKING BOUNDARIES OF CULTIVATED LAND. 2. CONSTRUCTION OF PALACES, TEMPLES, LAKES, DAMS. YOU USE DIFFERENT OBJECTS HAVING DIFFERENT SHAPES. 3

SHAPES OF SOME OBJECTS: 1. PEN/PENCIL/RULER- STRAIGHT 2. BANGLE/BALL- ROUND GEOMETRY IS THE BRANCH IN WHICH WE GET IDEAS ABOUT SHAPES OF OBJECTS AND STUDY ABOUT THEM. 4

POINT- THE FIRST BASIC SHAPE BY A SHARP TIP OF A PENCIL MARK A DOT ON THE PIECE OF PAPER. SHARPER THE TIP THINNER WILL BE THE DOT. THE POINT DETERMINES A LOCATION. 5

MODELS OF A POINT: POIN T THE SHARPENED END OF A PENCIL: : THE TIP OF A COMPASS: POINT 6

TO DISTINGUISH BETWEEN POINTS WE DENOTE EACH POINT BY A SINGLE CAPITAL LETTER. E. G, A, B, C ETC. THESE ARE READ AS POINTA, POINT B, POINT C ETC. 7

LINE SEGMENT –THE SECOND BASIC SHAPE FOLD A PIECE OF PAPER AND UNFOLD IT. DO YOU SEE A FOLD? THIS GIVES THE IDEA OF A LINE. A B A THREAD HELD WITHOUT A SLACK 8

MODELS OF A LINE SEGMENT: AN EDGE OF A BOX STRAIGHT LINE AN EDGE OF A POST CARD 9

THE SHORTEST ROUTE TO CONNECT TWO POINTS A AND B IS A STRAIGHT LINE. IT IS DENOTED BY AB OR BA. THE POINTS A AND B ARE CALLED END POINTS OF THE LINE SEGMENT. A LINE: IF A LINE SEGMENT AB IS EXTENDED BEYOND A IN ONE DIRECTION AND BEYOND B IN ANOTHER DIRECTION, WE GET A LINE. A B DENOTED BY l. WE CANNOT GET THE COMPLETE PICTURE OF A LINE BECAUSE IT EXTENDS INDEFINITELY IN BOTH DIRECTIONS. 10

HENCE IT IS NECESSARY TO FIX A LINE. THIS LINE CAN BE FIXED BY TWO POINTS. A B TWO POINTS DETERMINE A LINE. 11

INTERSECTING LINES P l 1 POINT OF INTERSECTION MODELS OF INTERSECTING LINES: l 2 ADJACENT EDGES OF A BOX ENGLISH ALPHABET X 12

TELL THE STUDENTS TO DISCUSS WHETHER: 1. TWO LINES INTERSECT IN MORE THAN ONE POINT? 2. MORE THAN TWO LINES INTERSECT IN ONE POINT. PARALLEL LINES: AB AND BD ARE INTERSECTING LINES A B C D AB AND CD ARE LINES WHICH DO NOT MEET. SUCH LINES ARE CALLED PARALLEL LINES. AC AND BD ARE ALSO SUCH PAIR OF LINES. 13

RAY SUN RAYS P A B SUN C IT HAS A STARTING POINT B. IT EXTENDS ENDLESSLY IN ONE DIRECTION. TO FIX THE RAY, A POINT P IS TAKEN ON ITS PATH. THE RAY IS DENOTED BY BP 14

CURVES 2 1 3 6 4 5 THE FIGURES DRAWN WITHOUT LIFTING THE PENCIL FROM THE PAPER AND WITHOUT THE USE OF A RULER ARE CURVES. 15

WHAT CAN YOU SAY ABOUT THE FIG. 6. IS IT A CURVE OR NOT ? WELL, IN EVERY DAY USAGE CURVE MEANS NOT STRAIGHT. HOWEVER IN MATHEMATICS, A STRAIGHT LINE IS A TYPE OF CURVE. WE OBSERVE THAT CURVES 4 AND 5 CROSS THEMSELVES WHILE OTHERS DO NOT. ALL THE CURVES EXCEPT 4 AND 5 ARE SIMPLE CURVES 16

SIMPLE CURVES ARE FURTHER SUBDIVIDED INTO OPEN CURVES (1) AND CLOSED CURVES (2) AND (3). POSITION IN A FIGURE: A COURT LINE IN A TENNIS COURT DIVIDES IT INTO 3 PARTS: INSIDE THE LINE; ON THE LINE AND OUTSIDE THE LINE. IT IS A CLOSED CURVE. THERE ARE 3 DISJOINT PARTS: 1. INTERIOR(INSIDE) OF THE CURVE. 2. BOUNDARY(ON) THE CURVE. 3. EXTERIOR(OUTSIDE) THE CURVE. 17

A B C A IS IN THE INTERIOR B IS ON THE BOUNDARY C IS IN THE EXTERIOR. 18

POLYGONS 1) ARE THEY CLOSED-YES ALL THE FIGURES ARE MADE OF 2) LINE SEGMENTS THESE ARE POLYGONS 19

SIDES, VERTICES AND DIAGONALS D GIVE A JUSTIFICATION TO CALL IT A POLYGON E C LINE SEGMENTS FORMING A POLYGON ARE CALLED ITS SIDES A B TELL THE STUDENTS TO NAME THE SIDES AB, BC, CD, DE AND EA THE MEETING POINT OF A PAIR OF SIDES IS CALLED THE VERTEX. SIDES AE AND ED MEET AT E SO E IS A VERTEX 20

SIMILARLY B, C, D ARE OTHER VERTICES. NAME THE SIDES THAT MEET AT THESE POINTS. THE SIDES AB AND BC ARE ADJACENT WHILE AE AND DC ARE NOT. THESE ARE THE SIDES WITH COMMON END POINT B. THE ENDPOINTS OF THE SAME SIDE OF THE POLYGON ARE CALLED ADJACENT VERTICES E AND D ARE ADJACENT. VERTICES A AND D ARE NOT. NAME THE OTHER ADJACENT VERTICES. 21

CONSIDER THE PAIRS OF VERTICES WHICH ARE NOT ADJACENT. THE JOINS OF THESE VERTICES ARE CALLED THE DIAGONALS OF THE POLYGON. D B & E ARE ADJACENT. SO THE JOIN OF B E AND E IS A DIAGONAL. C A CAN YOU IDENTIFY OTHER DIAGONALS? B 22

ANGLES ARE MADE WHEN CORNERS ARE FORMED. THIS FIGURE GIVES THE IDEA OF AN ANGLE AB AND AC ARE TWO RAYS WITH A C B A COMMON END POINT A. TWO RAYS WHICH START FROM A COMMON END POINT FORM AN ANGLE. THE COMMON END POINT IS CALLED THE VERTEX TWO RAYS AB AND AC ARE CALLED ARMS. 23

P WE ACTUALLY SHOW THE ANGLE AS O Q THE SMALL CURVE INDICATES IT. TO NAME THIS ANGLE WE CAN SIMPLY SAY THAT THIS IS ANGLE O. HOWEVER TO MAKE IT MORE SPECIFIC WE SAY IT ANGLE POQ. INTERIOR OF THE TRIANGLE. O 24

TRIANGLES A IT IS A THREE SIDED POLYGON. IT IS A POLYGON WITH LEAST NO. OF SIDES. B C CAN YOU IDENTIFY THE NUMBER OF SIDES AND NUMBER OF ANGLES IN THE FIGURE? YES IT HAS THREE SIDES AND THREE ANGLES. HOW MANY VERTICES DOES IT HAVE? JUSTIFY. THIS FIGURE HAS AN EXTERIOR AND AN INTERIOR. 25

Q R P P IS IN THE INTERIOR. Q IS ON THE TRIANGLE. R IS IN THE EXTERIOR. 26

QUADRILATERAL C A FOUR SIDED FIGURE IS CALLED A QUADRILATERAL. HOW MANY SIDES AND ANGLES DOES IT HAVE? NAME THEM. D B A HOW MANY VERTICES DOES IT HAVE? NAME THEM. IDENTIFY THE NON ADJACENT VERTICES AND DIAGONALS. 27

THE VERTICES ARE NAMED INTERIOR IN THE CYCLIC MANNER B ANTI-CLOCKWISE. ANGLE A AND ANGLE C ARE CALLED OPPOSITE ANGLES A AND B ARE ADJACENT ANGLES. LIST OTHER PAIRS OF ADJACENT ANGLES. 28

CIRCLE PLACE A BANGLE OR ANY ROUND SHAPE, TRACE AROUND TO GET CIRCULAR SHAPE TO DRAW A CIRCLE TAKE TWO STICKS AND CIRCULAR A PIECE OF THREAD. SHAPE DRIVE ONE STICK INTO THE GROUND (OA). PROPOSED BANGLE CENTRE OF THE CIRCLE. MAKE TWO LOOPS ONE LOOP AT EACH END OF THE THREAD. PLACE ONE LOOP AROUND THE STICK AT THE CENTRE AND THE OTHER AROUND THE OTHER STICK. KEEP STICKS VERTICAL. TRACE THE A STICK C STICK D O D A E R TH GROUND CIRCLE KEEPING THE STICK AT THE CENTRE FIXED. 29

CIRCLE IS A CLOSED CURVE WHICH IS NOT A POLYGON. IT HAS CERTAIN VERY SPECIAL PROPERTIES. MODEL OF A CIRCLE EQUIDISTANT FROM CENTRE 30

PARTS OF A CIRCLE A HERE IS A CIRCLE WITH CENTRE C. A, P, B, M ARE POINTS ON THE CIRCLE P C M CA = CB=CP=CM RADII OF A CIRCLE B PLURAL OF RADIUS PM IS THE DIAMETER OF THE CIRCLE. IS THE SIZE OF THE DIAMETER DOUBLE THE RADIUS? YES. SO DIAMETER IS DOUBLE THE RADIUS. 31

DIAMETER SE CI MI RC SE L CI MI E RC LE THE DIAMETER DIVIDES THE CIRCLE INTO TWO DALF CIRCLES CALLED SEMI CIRCLES 32

A P C PB IS THE CHORD CONNECTING M B TWO POINTS ON A CIRCLE. IS PM ALSO A CHORD? YES? P AN ARC IS A PORTION OF A CIRCLE. IF P AND Q ARE TWO POINTS WE GET THE ARC PQ. WE WRITE IT AS Q PQ. 33

P INTERIOR EXTERIOR Q R SECTOR OF A CIRCLE A REGION IN THE INTERIOR OF A CIRCLE ENCLOSED BY ARC ON ONE SIDE AND PAIR OF RADII. 34

ARC AB SEGMENT OF A CIRCLE A THE INTERIOR OF A CIRCLE ENCLOSED BY CHORD AND B THE ARC CHORD AB CIRCUMFERENCE THE LENGTH AROUND THE CIRCLE 35