Discussion Forum

In a  $\triangle XYZ$ , let x,y,z be the lengths of sides opposite to the angle  X,Y,Z respectively and 2s=x+y+z. If  $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the $\triangle XYZ$ is  $\frac{8\pi}{3}$ then

Answer:

Explanation:

Given a $\triangle XYZ$  ,where 2x=x+y+z

and   $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$

$\therefore$                       $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ 

=  $\frac{3x-(x+y+z)}{4+3+2}=\frac{s}{9}$

 or  $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}=\frac{s}{9}=\lambda(let)$

$\Rightarrow$    $s=9\lambda,s=4\lambda+x, s=3\lambda+y$ 

 and  $s=2\lambda+z$

$\therefore$    $s=9\lambda,x=5\lambda, y=6\lambda, z=7\lambda$

Now,    $\triangle= \sqrt{s(s-x)(s-y)(s-z)}$   [heron's formula]

$= \sqrt{9\lambda.4\lambda.3\lambda.2\lambda}=6\sqrt{6}\lambda^{2}$          .........(i)

 Also,  $\pi r^{2}=\frac{8\pi}{3}$

 $\Rightarrow$     $r^{3}=\frac{8}{3}$          .......(ii)

and   $R=\frac{xyz}{4\triangle}$

 $=\frac{(5\lambda)(6\lambda)(7\lambda)}{4.6.\sqrt{6}\lambda^{2}}=\frac{35\lambda}{4\sqrt{6}}$  .......(iii)

Now,   $r^{2}=\frac{8}{3}=\frac{\triangle^{2}}{S^{2}}=\frac{216\lambda^{2}}{81\lambda^{2}}$

 $\Rightarrow$    $\frac{8}{3}=\frac{8}{3}\lambda^{2}$               [from Eq. (ii)]

$\Rightarrow$       $\lambda=1$

 (a)      $\triangle XYZ=6\sqrt{6}\lambda^{2}=6\sqrt{6}$

 $\therefore$      Option (a) is correct.

(b) Radius of circumcircle 

     $(R)=\frac{35}{4\sqrt{6}}\lambda=\frac{35}{4\sqrt{6}}$

$\therefore$  Option (b) is incorrect.

(c) Since,  $r= 4R\sin\frac{X}{2}.\sin\frac{Y}{2}.\sin\frac{Z}{2}$

  $\Rightarrow\frac{2\sqrt{2}}{\sqrt{3}}=4.\frac{35}{4\sqrt{6}}\sin\frac{X}{2}\sin\frac{Y}{2}\sin\frac{Z}{2}$

   $\Rightarrow\frac{4}{35}=\sin\frac{X}{2}\sin\frac{Y}{2}\sin\frac{Z}{2}$

 $\therefore$     Option (c) is correct.

(d)  $\sin^{2}\left(\frac{X+Y}{2}\right)=\cos^{2}\left(\frac{Z}{2}\right)$ .  as

     $\frac{X+Y}{2}=90^{0}-\frac{Z}{2}=\frac{s(s-z)}{xy}=\frac{9\times 2}{5\times 6}=\frac{3}{5}$

$\therefore$   Option (d) is correct