Para encontrarmos o valor de m, realizaremos os cálculos abaixo:

\(\begin{align} & \text{ }AB\text{ }=\text{ }B-A=\text{ }\left( 2m-1,20 \right)-\left( m,4 \right)\text{ }=\text{ }\left( m-1,16 \right) \\ & \left| AB \right|\text{ }=~\surd \left( m-1 \right){}^\text{2}+256\text{ }=~\surd 265 \\ & \left( m-1 \right){}^\text{2}+256\text{ }=\text{ }265 \\ & m{}^\text{2}-2m+1+256\text{ }=\text{ }265 \\ & m{}^\text{2}-2m+257-265\text{ }=\text{ }0 \\ & m{}^\text{2}-2m-8=0 \\ & m'=-2 \\ & m''=4 \\ \end{align}\ \)

Portanto, teremos que m=-2 ou m=4.