If the mid-points of the sides of a triangle are (1, 1), (2, -3) and (3, 4),find its vertices.

Let the vertices of the triangle be A(x ₁ , y ₁ ), B(x ₂ , y ₂ ) and C(x ₃ , y ₃ ). The midpoints of the sides of a triangle are D(1, 1), E(2, -3) and F(3, 4).

We know that the midpoint of a line segment AB with A(x, y) and B(u, v) is

So, the midpoint of AB:

⇒ x ₁ + x ₂ = 2 ………………………..(i)
and y ₁ + y ₂ = 2 ………………………..(ii)

The midpoint of BC:

⇒ x ₂ + x ₃ = 4 ………………………..(iii)
and y ₂ + y ₃ = -6 ………………………..(iv)

The midpoint of AC:

⇒ x ₃ + x ₁ = 6 ………………………..(v)
and y ₃ + y ₁ = 8 ………………………..(vi)
Adding the equations (i), (iii), (v) we get,
2(x ₁ + x ₂ + x ₃ ) = 12
⇒ x ₁ + x ₂ + x ₃ = 6...............(vii)
Substitute eqn.(i) in (vii) then x ₃ = 4.
Substitute eqn.(iii) in (vii) then x ₁ = 2.
Substitute eqn.(v) in (vii) then x ₂ = 0.
Adding the equations (ii), (iv), (vi) we get,
2(y ₁ + y ₂ + y ₃ ) = 4
⇒ y ₁ + y ₂ + y ₃ = 2...............(viii)
Substitute eqn.(ii) in (viii) then y ₃ = 0.
Substitute eqn.(iv) in (viii) then y ₁ = 8.
Substitute eqn.(vi) in (viii) then y ₂ = -6.
∴ The vertices A(x ₁ , y ₁ ), B(x ₂ , y ₂ ) and C(x ₃ , y ₃ ) = A(2, 8), B(0, -6) and C(4, 0)