If the mid-points of the sides of a triangle PQR are A(-1, -3), B(2, 1) andC(4, 5), find the coordinates of P, Q and R.

SOLUTION:

Let the vertices of the triangle be P(x ₁ , y ₁ ), Q(x ₂ , y ₂ ) and R(x ₃ , y ₃ ). The midpoints of the sides of a triangle are A(-1, -3), B(2, 1) and C(4, 5).

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

So, the midpoint of PQ:

⇒ x ₁ + x ₂ = -2 ………………………..(i)
and y ₁ + y ₂ = -6………………………..(ii)

The midpoint of QR:

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

The midpoint of RP:

⇒ x ₃ + x ₁ = 8 ………………………..(v)
and y ₃ + y ₁ = 10 ………………………..(vi)
Adding the equations (i), (iii), (v) we get,
2(x ₁ + x ₂ + x ₃ ) = 10
⇒ x ₁ + x ₂ + x ₃ = 5...............(vii)
Substitute eqn.(i) in (vii) then x ₃ = 7.
Substitute eqn.(iii) in (vii) then x ₁ = 1.
Substitute eqn.(v) in (vii) then x ₂ = -3.
Adding the equations (ii), (iv), (vi) we get,
2(y ₁ + y ₂ + y ₃ ) = 6
⇒ y ₁ + y ₂ + y ₃ = 3...............(viii)
Substitute eqn.(ii) in (viii) then y ₃ = 9.
Substitute eqn.(iv) in (viii) then y ₁ = 1.
Substitute eqn.(vi) in (viii) then y ₂ = -7.
∴ The vertices = P(1, 1), Q(-3, -7) and R(7, 9)