Question

Show that the points P(2, 3, 5), Q(-4, 7, -7), R(-2, 1, -10) and S(4, -3, 2) are the vertices of a rectangle.

Answer 1

To prove: Points P, Q, R, S forms rectangle. 

Formula: The distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂) is given by

D = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Here, 

(x₁,y₁,z₁) = (2, 3, 5) 

(x₂,y₂,z₂) = (-4, 7, -7) 

(x₃,y₃,z₃) = (-2, 1, -10) 

(x₄,y₄,z₄) = (4, -3, 2)

Length PQ = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Length RS = \(\sqrt{(x_4-x_3)^2+(y_4-y_3)^2+(z_4-z_3)^2}\)

Length PR = \(\sqrt{(x_3-x_1)^2+(y_3-y_1)^2+(z_3-z_1)^2}\)

Here, PQ = RS which are opposite sides of polygon. 

QR = PS which are opposite sides of polygon. 

Also the diagonals PR = QS. 

Hence, the polygon is a rectangle.