Q) In a triangle ABC , a:b:c=1+x:1:1-x , where x element of left parenthesis 0 comma 1 right parenthesis space. space I f space angle A space equals space 90 degree space plus angle C space t h e n space x equals
left parenthesis A right parenthesis fraction numerator 1 over denominator square root of 6 end fraction space space space left parenthesis B right parenthesis fraction numerator 1 over denominator 2 square root of 3 end fraction space space space space left parenthesis c right parenthesis fraction numerator 1 over denominator square root of 7 end fraction space space left parenthesis D right parenthesis fraction numerator 1 over denominator 2 square root of 7 end fraction

Asked by araima2001 | 5th Apr, 2017, 05:13: AM

Expert Answer:

begin mathsize 16px style The space sides space of space straight a space triangle space are space in space AP space and space the space greatest space angle space exceeds space the space least space angle space by space 90 degree
then space sides space are space proportional space to space square root of 7 plus 1 colon square root of 7 colon square root of 7 minus 1.

Proof colon
Given space that space straight a greater than straight b greater than straight c space rightwards double arrow straight a plus straight c equals 2 straight b
By space sin space rule comma
straight a over sinA equals straight b over sinB equals straight c over sinC equals 2 straight R
Where space straight R space is space the space radius space of space circumcircle.
angle straight A equals 90 degree plus angle straight C. space Let comma space angle straight C space be space straight x.
rightwards double arrow fraction numerator straight a over denominator sin open parentheses 90 degree plus straight x close parentheses end fraction equals fraction numerator straight b over denominator sin open parentheses 90 degree minus 2 straight x close parentheses end fraction equals straight c over sinx equals 2 straight R
rightwards double arrow straight a equals 2 Rcosx comma space straight b equals 2 Rcos 2 straight x comma space straight c equals 2 Rsinx.
straight a plus straight c equals 2 straight b
rightwards double arrow cosx plus sinx equals 2 cos 2 straight x
space space space space space space space space space space space space space space space space space space space space space space space space equals space 2 open parentheses cos squared straight x minus sin squared straight x close parentheses space space space
space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 open parentheses cosx plus sinx close parentheses open parentheses cosx minus sinx close parentheses
90 degree minus 2 straight x greater than 0 comma space cos 2 straight x greater than 0
alos space cosx greater than 0 space and space sinx greater than 0 space
rightwards double arrow cosx plus sinx greater than 0
cosx plus sinx equals 2 open parentheses cosx plus sinx close parentheses open parentheses cosx minus sinx close parentheses
1 equals 2 open parentheses cosx minus sinx close parentheses
rightwards double arrow open parentheses cosx minus sinx close parentheses equals 1 half
rightwards double arrow open parentheses cosx minus sinx close parentheses squared equals open parentheses 1 half close parentheses squared
rightwards double arrow 1 minus sin space 2 straight x equals 1 fourth
rightwards double arrow sin space 2 straight x equals 3 over 4
rightwards double arrow cos space 2 straight x equals fraction numerator square root of 7 over denominator 4 end fraction
cos squared straight x equals fraction numerator 1 minus cos 2 straight x over denominator 2 end fraction equals open parentheses fraction numerator square root of 7 plus 1 over denominator 4 end fraction close parentheses squared
cosx equals fraction numerator square root of 7 plus 1 over denominator 4 end fraction
Put space it space in space the space fraction numerator straight a over denominator sin open parentheses 90 degree plus straight x close parentheses end fraction equals fraction numerator straight b over denominator sin open parentheses 90 degree minus 2 straight x close parentheses end fraction equals straight c over sinx equals 2 straight R
You space will space get space that space sides space are space in space square root of 7 plus 1 colon square root of 7 colon square root of 7 minus 1.
If space we space divide space it space by space square root of 7 space we space get comma
1 plus fraction numerator 1 over denominator square root of 7 end fraction colon 1 colon 1 minus fraction numerator 1 over denominator square root of 7 end fraction.
Hence comma space answer space is space fraction numerator 1 over denominator square root of 7 end fraction.
end style

Answered by Sneha shidid | 7th Apr, 2017, 10:43: AM