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Question
Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex.
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Solution
Vertices A and C are (3, –1, 2), (−1, 1, 2) respectively.
Coordinates of the midpoint P between A and C `((3 - 1)/2, (-1 + 1)/2, (2 + 2)/2)` = (1, 0, 2)
Let the coordinates of point D be (x, y, z) and the coordinates of point B be (1, 2, −4).
∴ Mid point of DB `(("x" + 1)/2, ("y" + 2)/2, ("z" - 4)/2)`
The diagonals of a parallelogram divide each other into 2 equal parts.
Therefore, `("x" + 1)/2 = 1, ("y" + 2)/2 = 0, ("z" - 4)/2 = 2`
Hence, the coordinates of point D are (1, –2, 8).
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