Question

If the mid-points of the sides of a triangle are $(1,1),(2,4)$ and $(3,5)$, then the area (in sq. units) of the triangle is

Answer 1

Let the mid-points be $P(1,1),Q(2,4)$ and $R(3,5)$
We know that,
Area of triangle formed by mid-points $=\frac{1}{4}\times$ Area of original triangle.


$\text{Area}\left(△PQR\right)=\frac{1}{4}\times \text{Area}\left(△ABC\right)$
Now, $\text{Area}\left(△PQR\right)=\frac{1}{2}\left|\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\\ x_3& y_3\\ x_1& y_1\end{array}\right|$
$\text{Area}\left(△PQR\right)=\frac{1}{2}\left|\begin{array}{cc}1& 1\\ 2& 4\\ 3& 5\\ 1& 1\end{array}\right|$
$\Rightarrow \text{Area}\left(△PQR\right)=\frac{1}{2}|(4-2)+(10-12)+(3-5)|$
$\Rightarrow \text{Area}\left(△PQR\right)=\frac{1}{2}|2|=1\therefore \text{Area}\left(△ABC\right)=4$