The mid-point of the sides of a triangle are (1, 5, – 1), (0, 4, – 2) and (2, 3, 4). Find its vertices. Also find the centroid of the triangle.
Given:
Mid-points of sides of triangle ABC are,
D(1,5,-1), E(0,4,-2) & F(2,3,4)
Let us assume that the vertices of the triangle are –
A(x1, y1, z1), B(x2, y2, z2) & C(x3, y3, z3).
Now, we know that E is the mid-point of AC, thus,
(x1 + x3/2, y1 + y3/2, z1 + z3/2) = (0,4,-2)
Thus,
C(x3, y3, z3) = C(-x1, 8 – y1, -4-z1)
Now, mid-point of AB, F is
(x1 + x2/2, y1 + y2/2, z1 + z2/2) = (2,3,4)
Now, B(x2, y2, z2) = B(4-x1, 6-y1, 8-z1)
Now, mid-point of BC, D is,
-x1 + 4 – x1/2 = 1,
8 – y1 + 6 – y1/2 = 5
& - 4 – z1 + 8 – z1/2 = -1
Thus, x1 = 1, y1 = 2 & z1 = 3
Thus, A = (1,2,3), B = (3,4,5) & C = (-1,6,-7)
Now, for centroid,
G = (1+3-1/3,2+4+6/3, 3+5-7/3)
= (1,4,1/3)