If the three consecutive vertices of a parallelogram be A \left(\right. 3 , 4 , - 1 \left.\right) , B \left(\right. 7 , 10 , - 3 \left.\right) and C \left(\right. 5 , - 2 , 7 \left.\right), find the fourth vertex D.

Engineering

Mathematics

Line of Greatest Slope

Question

If the three consecutive vertices of a parallelogram be

A (3, 4, - 1), B (7, 10, - 3)

and

C (5, - 2, 7)

, find the fourth vertex

D

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Given: Three vertices

A (3, 4, - 1), B (7, 10, - 3)

and

C (5, - 2, 7)

of a parallelogram ABCD.

To find: fourth vertex D.

Let the co-ordinates of the fourth vertex D be (x, y, z).

We know, the diagonals of the parallelogram, AC and BD bisect each other.

∴

Let P be their mid-point with co-ordinates (a, b, c).

a = \frac{3 + 5}{2} = \frac{8}{2} = 4

b = \frac{4 - 2}{2}, = \frac{2}{2} = 1

c = \frac{- 1 + 7}{2} = \frac{6}{2} = 3

∵

Mid-point of AC and BD are the same point P.

∴ \frac{x + 7}{2} = 4

\frac{y + 10}{2} = 1

\frac{z + (- 3)}{2} = 3

x = 1

∴ y = - 8

∴ z = 3

Hence, fourth vertex D is (1, -8, 3)

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