The coordinates of the vertices of ∆PQR are P(-2,5), Q(-1,1), and R(7,3). Determine whether ∆PQR is a right triangle. Show your work and explain your answer

Solution: Yes, triangle PQR is a right angle triangle.Explanation:It is given that P(-2,5), Q(-1,1) and R(7,3).Distance between two points [tex]A(x_1,y_1)[/tex] and [tex]B(x_2,y_2)[/tex] is given by he formula,[tex]AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Use this formula to find the length of sides.[tex]PQ=\sqrt{(1)^2+(-4)^2} =\sqrt{17}[/tex][tex]QR=\sqrt{(8)^2+(2)^2} =\sqrt{68}[/tex][tex]PR=\sqrt{(9)^2+(-2)^2} =\sqrt{85}[/tex]By pythagoras theorem a triangle is a right angle triangle if and only if the sum of squares of two small sides is equal to the square of the largest side.Since the greatest side is PR.[tex](PQ)^2+(QR)^2=(\sqrt{17})^2+(\sqrt{68})^2 \\(PQ)^2+(QR)^2=17+68\\(PQ)^2+(QR)^2=85\\(PQ)^2+(QR)^2=(PR)^2[/tex]Hence, the triangle PQR is a right angle triangle.