Question

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

Answer 1

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


$\text{Area}\left(△PQR\right)=\frac{1}{4}\times \text{Area}\left(△ABC\right)$
Now,
$\text{Area}\left(△PQR\right)=\frac{1}{2}\left|\begin{array}{cccc}1& 2& 3& 1\\ 1& 4& 5& 1\end{array}\right|\Rightarrow \text{Area}\left(△PQR\right)=\frac{1}{2}|(4-2)+(10-12)+(3-5)|\Rightarrow \text{Area}\left(△PQR\right)=\frac{1}{2}|2|=1\therefore \text{Area}\left(△ABC\right)=4$