Question

Find the area of the triangle $ABC$ with $A(1,-4)$ and mid points of sides through $A$ being $(2,-1)$ and $(0,-1)$

Answer 1

A(1,-4)

Considering,

$AB(2,-1)$

$AC(0,-1)$

Let $B(x_1,y_1)$

$(\frac{x_1+1}{2},\frac{y_1-4}{2})=(2,-1)$

$x_1=3,y_1=2$

$B(3,2)$

Let $C(x_2,y_2)$

$(\frac{x_2+1}{2},\frac{y_2-4}{2})=(0,-1)$

$x_2=-1,y_2=2$

$C(-1,2)$

Area$=\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$

$=\frac{1}{2}\left[3\right(2+4)+(-1\left)\right(-4-2)+1(2-2\left)\right]$

$=\frac{1}{2}[18+6+0]$

$=\frac{24}{2}$

Area of triangle $=12$ sq. units