Show that the □PQRS formed by P(2, 1), Q(-1, 3), R(-5, -3) and S(-2, -5) is a rectangle.

Answer: Opposite sides are equal and the diagonals are equal, so □PQRS is a rectangle.

Distance formula: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).

\(PQ=\sqrt{9+4}=\sqrt{13}\), \(QR=\sqrt{16+36}=\sqrt{52}\), \(RS=\sqrt{9+4}=\sqrt{13}\), \(SP=\sqrt{16+36}=\sqrt{52}\).

Opposite sides are equal, so PQRS is a parallelogram.

Diagonals: \(PR=\sqrt{49+16}=\sqrt{65}\), \(QS=\sqrt{1+64}=\sqrt{65}\).

A parallelogram with equal diagonals is a rectangle. (Also \(PQ^2+QR^2=13+52=65=PR^2\), so \(\angle Q=90^\circ\).)