Given: The mid-points of the sides of the triangle are P(-2, 3, 5), Q(4, -1, 7) and R(6, 5, 3).
To find: the coordinates of vertices A, B and C
Formula used:
Section Formula: A line AB is divided by C in m:n where A(x, y, z) and B(a, b, c).
The coordinates of C is given by
We know the mid-point divides side in the ratio of 1:1.
Therefore,
The coordinates of C is given by,
P(-2, 3, 5) is mid-point of A(x1, y1, z1) and B(x2, y2, z2)
Therefore,
Subtract (4), (5) and (6) from (7) separately:
Subtract (8), (9) and (10) from (11) separately:
Adding (12), (13) and (14):
⇒ z1 + z2 + z2 + z3 + z1 + z3 = 6 + 14 + 10
⇒ 2z1 + 2z2 + 2z3 = 30
⇒ 2(z1 + z2 + z3) = 30
⇒ z1 + z2 + z3 = 15………………………(15)
Subtract (8), (9) and (10) from (11) separately:
z1 + z2 + z3 – z1 – z2 = 15 – 10
⇒ z3 = 5
z1 + z2 + z3 – z2 – z3 = 15 – 14
⇒ z1 = 1
z1 + z2 + z3 – z1 – z3 = 15 – 6
⇒ z2 = 9
Hence, vertices of sides are A(0, 9, 1) B(-4,-3, 9) and C(12, 1, 5)
