Write the area of the triangle having vertices at `(a , b+c),\ (b , c+a),\ (c , a+b)dot`

Options

A. 0
B. `a+b+c`
C. `ab+bc+ca`
D. none of these

Answer

A

Explanation

Let `Delta` be the area of the triangole with vertices at `(a+b+c),(b+c+a) and (c,a+b)`. Then
`Delta` = absolute value of `(1)/(2)|{:(a,b+c,1),(b,c+a,1),(c,a+b,1):}|`
`rArr Delta` = Absolute value of `(1)/(2)|{:(a,b+c+,1),(b,b+c+a,1),(c,c+a+b,1):}|[underset(c_(2)rarrC_(2)rarr+c_(1))"Applying"]`
`rArr Delta` = Absolute value of `(1)/(2)(a+b+c)|{:(a,1,1),(b,1,1),(c,1,1):}|" " ["Taking" a+b+c "common from" C_(2)]`
`rArr Delta=0`