A) 0
B) ?1
C) 1
D) 2
Correct Answer: A
Solution :
\[(\mathbf{b}-\mathbf{a})\,.\,\left( \mathbf{c}-\frac{\mathbf{a}+\mathbf{b}}{2} \right)=\mathbf{b}\,.\,\mathbf{c}-\mathbf{b}\,.\,\left( \frac{\mathbf{a}+\mathbf{b}}{2} \right)\,-\mathbf{a}\,.\,\mathbf{c}+\frac{\mathbf{a}}{2}(\mathbf{a}+\mathbf{b})\] and \[|\mathbf{a}-\mathbf{c}|\,=\,|\mathbf{b}-\mathbf{c}|\] \[\Rightarrow \] \[\,|\mathbf{a}-\mathbf{c}{{|}^{2}}\,=\,|\mathbf{b}-\mathbf{c}{{|}^{2}}\] \ \[\mathbf{a}+\mathbf{b}=2\mathbf{c}\] Therefore, \[(\mathbf{b}-\mathbf{a}).\,\left( \mathbf{c}-\frac{\mathbf{a}+\mathbf{b}}{2} \right)=0.\]
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