Question

Show that the points P(1, 3, 4), Q(-1, 6, 10), R(-7, 4, 7) and S(-5, 1, 1) are the vertices of a rhombus.

Answer 1

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

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) = (1, 3, 4) 

(x₂,y₂,z₂) = (-1, 6, 10) 

(x₃,y₃,z₃) = (-7, 4, 7) 

(x₄,y₄,z₄) = (-5, 1, 1)

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

Length PS = \(\sqrt{(x_4-x_1)^2+(y_4-y_1)^2+(z_4-z_1)^2}\)

Length QS = \(\sqrt{(x_4-x_2)^2+(y_4-y_2)^2+(z_4-z_2)^2}\)

Here, PQ = RS =QR = PS . 

Also the diagonals PR ≠ QS. 

Hence, the polygon is a rhombus as all sides are equal and diagonals are not equal.