A) \[abc\]
B) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
C) \[ab+bc+ca\]
D) 0
Correct Answer: D
Solution :
Area\[=\frac{1}{2}\,\left| \,\begin{matrix} a & b+c & 1 \\ b & c+a & 1 \\ c & a+b & 1 \\ \end{matrix}\, \right|=\frac{1}{2}\,\left| \,\begin{matrix} a & a+b+c & 1 \\ b & a+b+c & 1 \\ c & a+b+c & 1 \\ \end{matrix}\, \right|\] (Applying \[{{C}_{2}}\to {{C}_{1}}+{{C}_{2}})\] \[=\frac{(a+b+c)}{2}\,\left| \,\begin{matrix} a & 1 & 1 \\ b & 1 & 1 \\ c & 1 & 1 \\ \end{matrix}\, \right|=0\]. Note : Students should remember this question as a fact.
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