5. Show that $\triangle P Q R$ with vertices $P(2,-5), Q(-2,3)$, and $R(4,1)$ is a right isosceles triangle. Support your answer with calculations and explain your reasoning. (6 marks)

Step 1 ：Calculate the side lengths of the triangle using the distance formula: \(PQ = \sqrt{(-2-2)^2+(3-(-5))^2} = \sqrt{16+64} = \sqrt{80} = 4\sqrt{5}\)

Step 2 ：Calculate the side lengths of the triangle using the distance formula: \(PR = \sqrt{(4-2)^2+(1-(-5))^2} = \sqrt{4+36} = \sqrt{40} = 2\sqrt{10}\)

Step 3 ：Calculate the side lengths of the triangle using the distance formula: \(QR = \sqrt{(-2-4)^2+(3-1)^2} = \sqrt{36+4} = \sqrt{40} = 2\sqrt{10}\)

Step 4 ：Since \(PQ = 4\sqrt{5}\), \(PR = QR = 2\sqrt{10}\), and \(PQ^2 = 80\), \(PR^2 + QR^2 = 40 + 40 = 80\)

Step 5 ：Since \(PQ^2 = PR^2 + QR^2\), by the Pythagorean theorem, \(\triangle PQR\) is a right triangle with \(\angle P\) being the right angle

Step 6 ：Since \(PR = QR\), \(\triangle PQR\) is an isosceles right triangle

Step 7 ：\(\boxed{\text{Triangle PQR is a right isosceles triangle}}\)