# The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C.

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The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C.

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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: ⇒ 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)