PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties. + + 80 0. The sides of the tringle re denoted y,, opposite to the ngles, nd respetively. Fig () 3. + + s The perimeter of the tringle.. THE SINE RULE: In tringle, prove tht R
Where, R is the irum rdius of the tringle. PROOF: Let S e the irumentre of the tringle. First prove tht R SE (I) : Let S e n ute ngle. Let P e ny point on the irle. Join P, Whih pss through S. Join P, so tht P 90 O. P (ngles in the sme segment). FROM Δ P, sin S P sin, R Fig ( ) SE (II) : Let e right ngle, ie., R 90 0 ( Fig 3 ), Then is the dimeter. sin S R Fig ( 3)
SE (III) : Let e n otuse ngle ( Fig ). join P, pssing through S. Join P, so tht P 90 0. Now 80 0 ( ) 80 0 ( Sine P is yli qudrilterl ) From Δ P, sin ( ) S Fig ( ) i.e.- sin (80 0 ) sin R R is true for ll vlues of. Similrly, we n prove, R, Thus, R. or r sin, r sin, r sin.
.3 THE OSINE RULE: In ny tringle, prove tht + - os + - os + - os D D ( Fig 5 ) ( Fig 6 ) ( Fig 7 ) PROOF: se ( I ) Let e n ute ngle ( Fig 5 ) Drw D. From Δ D D + D ( D ) + D
-. D + D + D -. D + ( Sine D + D ). D + ut from Δ D, os D os os..os + Or + os se ( II ) Let e right ngled, ie., 90 0 ( Fig 6 ) + ie., + ut + os + os 90 0 +, whih is true for right ngled tringle. se ( III ) Let e otuse ngle, ie., > 90 0 ( Fig 7 ). Drw D produed. From Δ D D + D ( + D ) + D
+. D + D + D +. D + ( Sine, D + D ).. D + ut from Δ D, os ( D ), ( D 80 0 ) os ( 80 0 ) - os, D os + ( os ) + + os Similrly, we n prove, + - os + - os Tips for Students: The ove formule n e written s: os, os, os, These results re useful in finding the osines of the ngle when numeril vlues of the sides re given. Logrithmi omputtion is not pplile sine the formule involve sum nd differene of terms. However, logrithmi method n e pplied t the end of simplifition to find ngle. THE PROJETION RULE:
In this rule, we show how, one side of tringle n e epressed in terms of other two sides. It is lled projetions rule. os + os, os + os, os + os. PROOF: Let e n ute ngle Drw D produed. In Fig ( i ) D + D [ NOTE : D is lled projetion of on nd D is the projetion of ] D + D () From Δ D, os D os os From Δ D, os D os os From () os + os os + os D (Fig 8) seii: When is right ngle, ie., 90 0 ( Fig 9 ). os, os () Sine 90 0, os os 90 0 0, We get, os 90 0 + os os (3)
( Fig 9 ) se III : When is otuse ngle ( Fig (iii) ) From Δ D, D - D () From Δ D, os, D os From Δ D os ( 80 0 ) os, D - os 80 0 From () os ( - os ) D ie., os + os os + os ( Fig 0 ) Similrly, os + os, os + os..5 THE LW OF TNGENTS: In ny Δ, Prove tht :. tn, tn
. tn tn, 3. tn tn PROOF: Using sine rule,.os os.sin os, os.tn tn tn tn sin g ot U Similrly, other two results n e proved y hnging sides nd ngles in yle order. EXPRESSIONS FOR HLF NGLES IN TERMS OF,, : In ny tringle, prove tht.6
. sin S S,. os S S, 3. tn S S SS PROOF: () We know tht sin os sin - ( Using osine rule for ) sin S S Sine + + s + s - + s -
sin s s ( Divide y ) sin ± S S If is ute, then sin is lwys positive. sin S S. sin + os + ( Using osine rule for ) sin sin s s Using + + s + s Dividing y, we get os s s ± S S Sine is ute, os is lwys positive nd therefore, os S S
3. tn sin os s s ss s s ss Similrly, we n show tht sin s s, os s s, tn s s ss sin s s, os s s, tn s s ss WORKED EXMPLES. If 3,, 5, in tringle, find the vlue of.) sin.) sin + os SOLUTION: Sine, + is stisfied y the given sides, they form right ngled tringle.
5 + 3 90 0, sin 90 0 sin sin 5 0 nd, sin + os sin( X 90 0 ) + os( X 90 0 ) sin 360 0 + os 360 0 0 +. Prove tht sin ( ) + sin ( ) + sin ( ) 0. SOLUTION: Now, sin( ) R sin. sin( ) ( Sine, R sin ) R sin.sin( ) Sine + + 80 0, + 80 0 sin( + ) sin R sin ( + ) sin ( ) R sin [ sin sin ] Similrly, sin ( ) R [ sin sin ] sin ( ) R [ sin sin ]
L. H. S. sin ( ) + sin ( ) + sin ( ) R [ sin sin ] + R [ sin sin ] + R [ sin sin ] R [ sin sin + sin sin + sin sin ] 3. Prove tht, in Δ, sin sin SOLUTION: L. H. S. sin sin X sin sin sin sin sin Sine sin ( + ) sin ( - ) sin sin sin sin sin R R R sin( + ) sin in Δ ( Using sine rule )
R R R. H. S.. Prove tht ( os os ) - SOLUTION: L. H. S. ( os os ) os os - (Using osine rule) - - R. H. S. 5. Prove tht sin + sin + sin 0 SOLUTION:
Now sin X sin os (Sine, sin sin os ) X R X (Using sine rule nd osine rule) R. Similrly, sin R. sin R. L. H. S R. + R. + R. R. [ ( ) + ( ) + ( ) ] R. [ 0 ] R. H. S 6. Find the gretest side of the tringle, whose sides re + +, +,. SOLUTION: Let + +, +,
Then, is the gretest side. Therefore is the gretest ngle. os 3 3 os 3 3 - os60 0 os( 80 0 60 0 ) os0 0 0 0 Therefore, the gretest ngle is 0 0 7. If sin + sin sin in Δ, Prove tht either 90 0 or 90 0. SOLUTION: sin + sin sin sin.os sin Using sin + sin D sin D. os D sin ( + ). os ( ) sin os sin ( + ) Sin( 80 0 ) sin
sin os ( ) sin os Dividing y sin oth sides, we get, os( ) os lso, os( ) os - [ Sine, os( - ) os ] ± When, + ut, + + 80 0, gives - 80 0, i.e., 80 0, 90 0 When -, + + + 80 0, gives Therefore, tringle is right ngled tringle. + 80 0, i.e., 80 0, 90 0 os os 8. Prove tht - - SOLUTION: os os L. H. S - (Using os sin ) sin sin - sin - - sin +
- - sin + sin - - R + R Sine, sin R sin R - R. H. S