Question

Question 1 Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) andC (–1, 1, 2). Find the coordinates of the fourth vertex.

Class X1 - Maths -Introduction to Three Dimensional Geometry Page 278

Answer 1

so the coordinate of the fourth vertix is (1,-2,8)

Answer 2

Let ABCD is a parallelogram and co-ordinate of point D ≡ (α, β, γ )
see attachment ,
Here AC and BD are diagonals intersect at P .

We know,
∴ co-ordinates of midpoint of BD = co-ordinates of midpoint of AC
Use midpoint formula ,
(x, y, z) ≡ { (x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2 }

so,
co-ordinates of midpoint of BD = {(α + 1)/2, ( β + 2)/2, (γ -4)/2 }
co-ordinates of midpoint of AC = {(3 -1)/2, ( -1 + 1)/2, (2 + 2)/2 }
here,
(α + 1)/2 = (3 -1)/2
α +1 = 2
α = 1

(β + 2)/2 = (-1 + 1)/2
β +2 = 0
β = -2

(γ - 4)/2 = (2 + 2)/2
γ -4 = 4
γ = 8

Hence, point D Ξ ( 1 , -2 , 8 )