Question

The mid-points of the sides BC, CA and AB of \(\Delta \mathrm{ABC}\) are \(\mathrm{D}(2, 1), \mathrm{E}(-1, -3)\) and \(\mathrm{F}(4, 5)\) respectively. Find the co-ordinates of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\).

Answer 1

Let $$A(x_1, y_1),\ B(x_2, y_2),\ C(x_3, y_3)$$.

D is the mid-point of BC: $$x_2 + x_3 = 4, y_2 + y_3 = 2$$

E is the mid-point of CA: $$x_3 + x_1 = -2, y_3 + y_1 = -6$$

F is the mid-point of AB: $$x_1 + x_2 = 8, y_1 + y_2 = 10$$

Adding all three equations of x:

$$2(x_1 + x_2 + x_3) = 4 - 2 + 8 = 10$$

$$\Rightarrow x_1 + x_2 + x_3 = 5$$

Subtracting the mid-point equations one by one:

$$x_3 = 5 - 8 = -3$$

$$x_1 = 5 - 4 = 1$$

$$x_2 = 5 - (-2) = 7$$

Adding all three equations of y:

$$2(y_1 + y_2 + y_3) = 2 - 6 + 10 = 6$$

$$\Rightarrow y_1 + y_2 + y_3 = 3$$

Subtracting: $$y_3 = 3 - 10 = -7$$

$$y_1 = 3 - 2 = 1$$

$$y_2 = 3 - (-6) = 9$$

Hence $$A(1,1),\ B(7,9)$$ and $$C(-3,-7)$$.

Check: mid-point of $$BC = (2, 1)$$, of $$CA = (-1, -3)$$ and of $$AB = (4, 5)$$ – as given.