Question

The co-ordinate of the mid-points of the sides of a triangle ABC are $D(2,1),E(5,3)$ and $F(3,7)$. Find the lengths and equations of its sides.

Answer 1

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

$4=x_1+x_2..........\left(i\right)$ and $2=y_1+y_2.....\left(iv\right)$

$10=x_1+x_3..........\left(ii\right)$ and $6=y_1+y_3.....\left(v\right)$

$6=x_2+x_3............\left(iii\right)$ and $14=y_2+y_3.....\left(vi\right)$

Subtracting $\left(iii\right)$ from $\left(ii\right)$,we get

$4=x_1-x_2$

Solving it with $\left(i\right)$, we get

$x_1=4,x_2=0,x_3=6$

Similarly,

$y_1=-3,y_2=5,y_3=9$

$AB=\sqrt{16+64}$ and equation, $(y+3)=\frac{5+3}{0-4}(x-4)$

$AB=\sqrt{80}$ and $-y-3=2x-8⟹y=-2x+5$

$AC=\sqrt{2^2+{12}^2}$ and equation $(y+3)=\frac{9+3}{6-4}(x-4)$

$AC=\sqrt{148}$ and $y+3=6x-24$

$BC=\sqrt{60}$ and $3y-15=2x$