Given; The mid-point of the sides of a triangle are (1, 5, – 1), (0, 4, – 2) and (2, 3, 4).
Let he vertices be A(x1, y1, z1), B(x2, y2, z2) and A(x3, y3, z3) respectively.
∴ x2 = −1,
y2 = 6,
z2 = −7.
∴ x1 = 2 − x2 = 3,
y1 = 10 − y2 = 4,
z1 = −4 – z2 = 5.
∴ x3 = – x2 = 1,
y3 = 8 – y2 = 2,
z3 = −4 − z2 = 3.
∴ A(−1, 6, −7), B(3, 4, 5), C(1, 2, 3) are the required vertices.
Centroid of a triangle is given by the average of the coordinates of its vertices or midpoint of sides.