a,b,c are the lengths of three sides of triangle ABC. If a, b c are releated by the releation `a^(2)+b^(2)+c^(2)`=`ab+bc+ca`, then the value of

Options

A. `(3)/(4)`
B. `(3)/(2)`
C. `(3sqrt3)/(2)`
D. `(9)/(4)`

Answer

d

Explanation

` a^(2)+b^(2)+c^(2)=ab+bc+ca`
or `a^(2)+b^(2)+c^(2)-ab-bc-ca=0`
or `2a^(2)+2b^(2)+2c^(2)-2ab-2bc-2ca=0`
or `a^(2)+b^(2)-2ab+b^(2)+c^(2)-2bc+c^(2)+a^(2)-2ca=0`
` (a-b)^(2)+(b-c)^(2)+(c-a)^(2)=0`
`therefore `a=b=c`
DeltaABC=` equllateral`Delta`
`therefore angleA=angleB=angleC=60^(@)`
So,`sin^(2)A+sin^(2)B+sin^(2)C`
`=sin^(2)60^(@)+sin^(2)60^(@)+sin^(2)60^(@)`
`=3((sqrt(3))/(2))^(2)`
`(9)/(4)`