ON CONSTRUTION

AYUSH VASHISTHA

SUPPOSE A LINE SEGMENT AND YOU HAVE TO DIVIDE IT IN A GIVEN RATIO,SAY 3:2 AND YOU MAY DO IT BY MEASURING

THE LENGTH AND THEN MARKING A POINT ON IT THAT DIVIDES IT IN THE GIVEN

RATIO.BUT SUPPOSE YOU DO NOT HAVE ANY WAY OF MEASURING IT

PRECISELY.HOW WOULD YOU FIND THE POINT? WE GIVE BELOW THE WAYS FOR

FINDING SUCH A POINT.

TO DIVIDE A LINE SEGMENT IN A GIVEN RATIO:-

STEPS OF CONSTRUTION:-

DRAW A LINE SEGMENT OF 7.6cm.

DRAW AN ANGLE OF 6O DEGREE FROM BOTH SIDES .

FROM ONE SIDE DIVIDE LINE INTO 5 EQUAL PARTS AND OTHER INTO 8 EQUAL PARTS.

NOW JOIN 5th PART AND 8th PART OF OTHER LINE.

FROM WHERE IT CUTS THE MAIN LINE IT IS DIVIDE INTO RATIO OF 5:8.

A

THIS CONSTRUTION INVOLVES TWO DIFFERENT SITUATIONS.IN ONE THE TRIANGLE TO BE

CONSTRUTED IS SMALLER AND IN OTHER IT IS LARGER THAN GIVEN TRIANGLE. HERE THE SCALE FACTOR MEANS THE RATIO OF SIDES OF TRIANGLE

TO BE CONSTRUCTED WITH THE CORRESPONDING SIDES OF GIVEN TRIANGLE. LET US TAKE THE

EXAMPLES FOR UNDERSANDING THE CONSTRUCTION INVOLVED.THE SAME METHODS

WOULD APPLY FOR GENERAL CASE ALSO.

CONSTRUCT A TRIANGLE SIMILAR TO GIVEN TRIANGLE ABC WITH ITS SIDES EQUAL TO 3:4 OF THE CORRESPONDING SIDES OF TRIANGLE ABC.

STEPS OF CONSTRUCTION:-

1)DRAW ANY RAY BX MAKING AN ACUTE ANGLE WITH BC ON THE SIDE OPPOSITETO THE VERTEX A.

2)LOCATE 4 POINTS B1,B2,B3 AND B4ON BX SO THAT BB1=B1B2=B2B3=B3B4.

3) JOIN B4C AND DRAW A LINE THROUGH B3 PARALLEL TO B4C TO INTERSECT BC AT C`.

4)DRAW A LINESEGMENT C` PARALLEL TO THE LINE CA TO INTERSECT BA AT A`

THEN TRIANGLE A`BC` IS THE REQUIRED TRIANGLE.

BB1 B2 B3 B4

C`

LET US CONSDER AN ACTIVITY FOR THIS:-

OBJECTIVETO CONSTRUCT A PAIR OF TANGENT TO A

GIVEN CIRCLE FROM A GIVEN EXTERNAL POINT USING COMPASS AND STRAIGHT EDGE ONLY.

KNOWLEDGE BEFOREHAND1)DRAWING A CIRCLE

2)TO LOCATE THE MID POINT OF A LINE SEGMENT .

MATERIAL NEEDED(1)COMPASS (2)HB PENCIL (3)DRAWING

SHEET (4)STRAIGHT EDGE(5)SKETCH PEN.

METHOD1)TAKE A CHART PAPER AND DRAW A CIRCLE OF GIVEN RADIUS,SAY 2cm,USING COMPASS.

NAME THE CENTRE AS O. [fig.(a).]

FIG.(A)

.O

2)MARK A GIVEN POINT P OUTSIDE THE CIRCLE AND JOIN P AS SHOWN IN FIGURE (b).

P

FIG.[B]

3)TAKING THE RADIUS MORE THAN HALF OF OP IN COMPASS AND CENTRE AS P ,DRAW TWO

ARCS ON EITHER SIDE OF PO AS SHOWN IN FIG.(c).

O

4)TAKING SOME RADIUS AND CENTRE AT O,REPEAT THE PROCESS IN STEP 3 INTERESTING PREVIOUS ARCS AT Q AND R .

(5) JOIN QR TO INTERSECT PO AT S AS SHOWN IN FIG.(D).

P

(6) TAKING PS or SO AS RADIUS ANDS AS CENTRE , DRAW A CIRCLE WHICH INTERSECTS THE CIRCLE HAVING CENTRE O IN M AND N AND JOIN PM AND PN AS SHOWN IN FIGURE(E).

CONCLUSION

PM AND PN ARE REQUIRED TANGENTS.

P

M

N