Question

Find the area of the triangle formed by joining the mid-point of the sides of the triangle whose vertices are $(0,-1),(2,1)$ and $(0,3)$. Find the ratio of this area to the area of the given triangle.

Answer 1

since $D,E$ and $F$ are the midpoints of $AB$, $BC$ and $CA$ of triangle respectively

coordinate of $D$$=(\frac{0+2}{2},\frac{-1+1}{2})=(1,0)$

coordinate of $E$$=(\frac{2+0}{2},\frac{1+3}{2})=(1,2)$

coordinate of $F$$=(\frac{0+0}{2},\frac{-1+3}{2})=(0,1)$

$\therefore ar\left(△DEF\right)=\frac{1}{2}\left[1\right(2-1)+1(1-0)+0(0-2\left)\right]=1sq.unitandar\left(△ABC\right)=\frac{1}{2}\left[0\right(1-3)+2(3+1)+0(-1-1\left)\right]=4sq.unit\frac{ar\left(△DEF\right)}{ar\left(△ABC\right)}=\frac{1}{4}\Rightarrow ar\left(△DEF\right):ar\left(△ABC\right)=1:4$