IIT JEE , AIEEE Forum - Mathematics , Trignometry

deraj_pv

hi!
my question is

If the lengths of sides of a,b,c of a triangle satisfy
2(bc²+ca²+ab²)=b² c+c²a+a²b+3abc ......then the triangle is a............?

please send me the solution.

puneet

Can you pls make the question a bit clear ... the powers are on which alphabet are not clear use the toolbar available to type in the expression. Once ur question is clear i will answer it .

cheers

deraj_pv

i have changed the question .....plz...answer my question

polymath

Equilateral.

edison

If the lengths of sides of a,b,c of a triangle satisfy
2(bc²+ca²+ab²)=b² c+c²a+a²b+3abc

or bc(2c-b) + ca(2a - c) + ab(2b - a) = 3abc

or bc(2c-b) -abc + ca(2a - c) -abc + ab(2b - a) - abc = 0

or bc(2c-b-a) + ca(2a - c - b) + ab(2b - a - c) = 0

so, (2c-b-a) = 0, (2a - c - b) = 0 and (2b - a - c) = 0

which further implies

2c = a + b

2a = c + b

2b = a + c

solving these three equations we obtain a = b = c

hence the triangle is equilateral

NOTE: since the equation is symmetric in a, b, and c. thus without deriving even we can conclude that the triangle is symmetric.

deraj_pv

thank u every much for solving my answer

ayshwarya

hey frnds do u suffer like dis 1stly ok frnd listen here if a condition is given 1ts v must check veder it is equilateral or not by putting a=b=c=some value [easy 2 calculate] if it isnt equilateral den proceed 2 rt angled triangle by applying magic no [but most of problems r of eqltral only]

rate me if u like it plz

ravi00

in this sum

there are 3 cases;

1.equi

2.iso

3.scalene

1st case:

leta=b=c=1

2(1+1+1)=1+1+1+3(1)

6=6.

its an equi triangle.

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