The value of [tex]y[/tex] will be [tex]4\sqrt{3}[/tex] .
Triangle is a three sides two dimensional surface having three angles.
We have,
[tex]\triangle MRN[/tex] , [tex]\triangle RSM[/tex] and [tex]\triangle RSN[/tex],
And, [tex]NS =6,\ SM=2,\ RM=x,\ NR=y,\ RS=z[/tex]
Now,
We have right angle [tex]\triangle MRN[/tex],
Using Pythagoras theorem,
[tex]x^2+y^2=8^2[/tex]
[tex]x^2+y^2+=64\[/tex] [tex].....(i)[/tex]
Now, We have right angle [tex]\triangle RSN[/tex],
Again Using Pythagoras theorem,
[tex]z^2+6^2=y^2[/tex]
⇒[tex]z^2=y^2-36[/tex] [tex].....(ii)[/tex]
Now, We have right angle [tex]\triangle RSM[/tex],
Again Using Pythagoras theorem,
[tex]z^2+2^2=y^2[/tex]
⇒[tex]z^2=x^2-4[/tex] [tex].....(iii)[/tex]
Now comparing equation [tex](ii)[/tex] and equation [tex](iii)[/tex],
We get,
[tex]x^2-4=y^2-36[/tex]
We get,
[tex]x^2-y^2=-36+4[/tex]
[tex]x^2-y^2=-32[/tex] [tex].....(iv)[/tex]
Now,
Adding equation [tex](i)[/tex] and equation [tex](iv)[/tex], i.e.
[tex]x^2+y^2+=64[/tex]
[tex]x^2-y^2=-32[/tex]
We get,
[tex]2y^2=96[/tex]
[tex]y^2=48[/tex]
[tex]y=\sqrt{48}[/tex]
[tex]y=4\sqrt{3}[/tex]
So, the value of [tex]y[/tex] is [tex]4\sqrt{3}[/tex]
Hence, we can say that the value of [tex]y[/tex] will be [tex]4\sqrt{3}[/tex] which is given in option (b).
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