check the picture below.
now, clearly the longest side is MP, and for a triangle to be an isosceles, it has to have two equal sides, so two sides must be twins.
since MP is the longest one, that only leaves out NP and NM, are they twins?
[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\
\begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
% (a,b)
&N&(~ 7 &,& -2~)
% (c,d)
&P&(~ 2 &,& 10~)
\end{array}\\\\\\
% distance value
d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}
\\\\\\
NP=\sqrt{[2-7]^2+[10-(-2)]^2}\implies NP=\sqrt{(2-7)^2+(10+2)^2}
\\\\\\
NP=\sqrt{(-5)^2+12^2}\implies NP=\sqrt{25+144}\implies \boxed{NP=13}[/tex]
and now let's check NM
[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\
\begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
% (a,b)
&N&(~ 7 &,& -2~)
% (c,d)
&M&(~ -5 &,& -7~)
\end{array}~~~
\\\\\\
NM=\sqrt{[-5-7]^2+[-7-(-2)]^2}\\\\\\ NM=\sqrt{(-5-7)^2+(-7+2)^2}
\\\\\\
NM=\sqrt{(-12)^2+(-5)^2}\implies NM=\sqrt{144+25}\implies \boxed{NM=13}[/tex]
