Answer :
First, let's check if all segments have the same length, calculating the distance between the points using the formula:
[tex]d=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]So we have:
[tex]\begin{gathered} AB\colon \\ d=\sqrt[]{(-1-4)^2+(4-2)^2}=\sqrt[]{25+4}=\sqrt[]{29} \\ \\ BC\colon \\ d=\sqrt[]{(-3-(-1))^2+(-1-4)^2}=\sqrt[]{4+25}=\sqrt[]{29} \\ \\ CD\colon \\ d=\sqrt[]{(2-(-3))^2+(-3-(-1))^2_{}_{}}=\sqrt[]{25+4}=\sqrt[]{29} \\ \\ AD\colon \\ d=\sqrt[]{(4-2)^2+(-3-2)^2}=\sqrt[]{4+25_{}}=\sqrt[]{29} \end{gathered}[/tex]Now, we need to check the slopes of each segment. The adjacent sides need to be perpendicular, so their slopes need to have the relation:
[tex]m_2=-\frac{1}{m_1}[/tex]Calculating the slopes with the formula below, we have:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \\ AB\colon \\ m=\frac{-1-4}{4-2}=-\frac{5}{2} \\ \\ BC\colon \\ m=\frac{-3-(-1)_{}}{-1-4}=\frac{2}{5} \\ \\ CD\colon \\ m=\frac{2-(-3)}{-3-(-1)}=-\frac{5}{2} \\ \\ AD\colon \\ m=\frac{4-2}{2-(-3)}=\frac{2}{5} \end{gathered}[/tex]So all adjacent sides are perpendicular.
All sides have the same length and all adjacent sides are perpendicular, therefore ABCD is a square.