Question

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

Answer 1

Consider the diagram shown below. Let $A,B,C$ be the vertices of the triangle and $D,E,F$ be the mid-points of sides $AB,BC,AC$, respectively.

Therefore,

Coordinates of $A=(2+3-5,1+7-3)=(0,5)$

Coordinates of $B=(2+5-3,1+3-7)=(4,-3)$

Coordinates of $C=(5+3-2,3+7-1)=(6,9)$

Therefore,

Length of $AB=\sqrt{(4-0{)}^2+(-3-5{)}^2}=4\sqrt{5}$

Length of $BC=\sqrt{(6-4{)}^2+(9+3{)}^2}=2\sqrt{37}$

Length of $AC=\sqrt{(6-0{)}^2+(9-5{)}^2}=2\sqrt{13}$