**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):

⇒ z_{1 }+ z_{2} + z_{2 }+ z_{3} + z_{1} + z_{3} = 6 + 14 + 10

⇒ 2z_{1} + 2z_{2} + 2z_{3} = 30

⇒ 2(z_{1} + z_{2} + z_{3}) = 30

⇒ z_{1} + z_{2} + z_{3} = 15………………………(15)

Subtract (8), (9) and (10) from (11) separately:

z_{1 }+ z_{2} + z_{3} – z_{1} – z_{2 }= 15 – 10

⇒ z_{3} = 5

z_{1} + z_{2} + z_{3} – z_{2} – z_{3} = 15 – 14

⇒ z_{1} = 1

z_{1} + z_{2} + z_{3} – z_{1} – z_{3} = 15 – 6

⇒ z_{2} = 9

Hence, vertices of sides are A(0, 9, 1) B(-4,-3, 9) and C(12, 1, 5)