Đáp án:

a) ABC là 1 tam giác nên:

$\begin{array}{l}
\left| {AC - AB} \right| < BC < \left| {AC + AB} \right|\\
 \Rightarrow \left( {6 - 1} \right) < BC < 6 + 1\\
 \Rightarrow 5 < BC < 7\\
 \Rightarrow BC = 6\left( {cm} \right)\\
b){C_{ABC}} = AB + AC + BC = 1 + 6 + 6 = 13\left( {cm} \right)\\
c)\\
Do:\left\{ \begin{array}{l}
MA + MB > AB\\
MB + MC > BC\\
MC + MA > AC
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
MA + MB > 1\\
MB + MC > 6\\
MC + MA > 6
\end{array} \right.\\
 \Rightarrow 2\left( {MA + MB + MC} \right) > 1 + 6 + 6\\
 \Rightarrow 2\left( {MA + MB + MC} \right) > 13\\
 \Rightarrow \left( {MA + MB + MC} \right) > \frac{{13}}{2}\\
 \Rightarrow \left( {MA + MB + MC} \right) > 6,5
\end{array}$

Vậy MA + MB + MC > 6,5