Question

PQRS is a parallelogram. Given that three of the vertices are P(6,-2,4), Q(2,4,-8), and R(-2,2,4), determine the coordinates of vertex S.

The vertex and vertices are the same thing but I do not know how to find the fourth point using the distance formula. Can someone please help me?

Answer 1

S = (2, -4, 16)

Explanation:

You don't need the distance formula to find the fourth vertex. The midpoint of the diagonals is the same for both diagonals, so ...

(P+R)/2 = (Q+S)/2 . . . . midpoints are the same

P + R = Q + S . . . . . . . .multiply by 2

P + R - Q = S . . . . . . . .subtract Q to get an equation for S

S = (6, -2, 4) +(-2, 2, 4) -(2, 4, -8)

= (6 -2 -2, -2 +2 -4, 4 +4 +8)

S = (2, -4, 16)