The co-ordinates of mid-point s of sides of a triangle are (4, 3), (–2, 7) and (0, 11). Let us calculate the co-ordinates of its vertices.

Let A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) be the vertices of triangle and D(4, 3) be the mid-point of AB, E(–2, 7) be the mid-point of BC and F(0, 11) be the mid-point of AC.

And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x₁, y₁) and (x₂, y₂) is

⇒ x₁ + x₂ = 8, ………(1)

y₁ + y₂ = 6, ………(2)

x₂ + x₃ = – 4, ……..(3)

y₂ + y₃ = 14, …….(4)

x₁ + x₃ = 0, ……..(5)

y₁ + y₃ = 22 ………(6)

Now, adding (1) and (3), we get –

x₁ + 2x₂ + x₃ = 4…..(7)

Now, subtracting (5) from (7), we get –

2x₂ = 4

⇒ x₂ = 2

Now, putting this in (3), we get –

2 + x₃ = – 4

⇒ x₃ = – 6

Now, putting this in (5), we get –

x₁ – 6 = 0

x₁ = 6

And now, adding (2) and (4), we get –

y₁ + 2y₂ + y₃ = 20…..(8)

Now, subtracting (6) from (8), we get –

2y₂ = – 2

⇒ y₂ = – 2

Now, putting this in (4), we get –

– 2 + y₃ = 14

⇒ y₃ = 16

Now, putting this in (6), we get –

y₁ + 16 = 22

y₁ = 6

∴, A(x₁, y₁) = (6, 7), B(x₂, y₂) = (2, – 1), C(x₃, y₃) = (– 6, 15)