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In a triangle XYZ, let x, y, z be the le...

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and 2s = x + y + z. If `(s-x)/4=(s-y)/3=(s-z)/2` of incircle of the triangle XYZ is `(8pi)/3`

A

are of the `DeltaXYZ` is `6sqrt6`

B

the radius of circumicircle of the `DeltaZYZ" is "(35)/6sqrt6`

C

`sin""X/2sin""Y/2sin""Z/2=4/35`

D

`sin^(2)((X+Y)/2)=3/5`

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
Given a `DeltaXYS,` where 2s=x+y+z
`"and "(epsi-x)/4=(epsi-y)/3=(epsi-z)/2`

`:." "(s-x)/4=(s-y)/3=(s-z)/2`
`=(3s-(x+y+z))/(4+3+2)=s/9`
`"or "(s-x)/4=(s-y)/3=(s-z)/2=s/9=lamda("let")`
`rArr" "s=9lamda,s=4lamda+x,s=3lamda+y`
`"and "s=2lamda+2`
`:." "s=9lamda, x=5lamda, y=6lamda, z=7lamda`
`"Now, "Delta=sqrt(s(s-x)(s-y)(z-z))`
[Heron's formula]
`=-sqrt(9lamda" "4lamda" "3lamda" "2lamda)=6sqrt6lamda^(2)" "....(i)`
`"Also"" "pir^(2)=8/3`
`rArr" "r^(2)=8/3" ...."(ii)`
`"and "R=(xyz)/(4Delta)=((5lamda)(6lamda)(7lamda))/(6" "6sqrt6lamda^(2))=(35lamda)/(4sqrt6)" ....."(iii)`
`"Now, "r^(2)=8/3=Delta^(2)/S^(2)=(216lamda^(4))/(81lamda^(2))`
`rArr" "8/3=8/3lamda^(2)" "["form Eq. (ii)]"`
`rArr" "lamda=1`
`(a) DeltaXYZ=6sqrt6lamda^(2)=6sqrt6`
`:.`Option (a) correct.
`"(b) Radius of circumcircle (R)"=35/(4sqrt6)lamda=35/(4sqrt6)`
`:.` Option a(a) is correct.
`"(C) Since, "r=4Rsin""X/2""sin""Y/2""sin""Z/2`
`rArr" "4/35=sin""X/2""sin""Y/2""sin""Z/2`
`:." Option (c) is correct."`
`"(d) sin"^(2)((X+Y)/2)=cos^(2)(Z/2)`
`"as"(X+Y)/2=90^(@)-Z/2=(s(s-z))/(xy)=(9xx2)/(5xx6)=3/5`
`:.` Option (d) is correct.
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