If the mid–points of the sides of a triangle are (1, 2), (0, –1) and (2, –1), then find the coordinates of the vertices of the triangle.

Let A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) be the vertices of the given triangle

and D (1, 2), E (2, –1) and F (0, –1) be the mid points of the sides BC, CA and AB respectively.

Using the mid–point formula,

=(0, –1)

⇒ x₁ +x₂ = 0 and y₁ +y₂ = –2 ……(i)

=(2, –1)

⇒ x₁ +x₃ = 4 and y₁ +y₃ = –2 ……(ii)

=(1, 2)

⇒ x₂ +x₃ = 2 and y₂ +y₃ = 4 ……(iii)

Adding all the equations obtained for x coordinates and y coordinates respectively, we get

2(x₁ + x₂ + x₃) = 6

and 2(y₁ + y₂ + y₃) = 0

⇒ x₁ + x₂ + x₃ = 3 and y₁ + y₂ + y₃ = 0 –––(iv)

From eq (i) and (iv), we get x₃ = 3 , y₃ = 2 , i.e. C (3, 2)

From eq (ii) and (iv), we get x₂ = –1 , y₂ = 2, i.e. B (–1, 2)

From eq (iii) and (iv), we get x₁ = 1 , y₃ = –4, i.e. A (1, –4)

∴ The coordinates of the vertices of the triangle are A (1, –4), B (–1, 2) and C (3, 2).