In $ riangle \mathrm{ABC}$, if $\mathrm{A}=(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively, then the area of $ riangle \mathrm{ABC}$ is

Question

In $	riangle \mathrm{ABC}$, if $\mathrm{A}=(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively, then the area of $	riangle \mathrm{ABC}$ is

Aicharya · Accepted Answer

$\mathrm{OC} \& \mathrm{OB}$ the equations of medians respectively and point $\mathrm{O}$ is the centroid. $\because$ Intersection OC \& OB is centroid. Solving $x+y=5 \& x=4$, we get the centroid as $(4,1)$. Let the coordinate of point $C=(4, a)$ and coordinate of point $B=(b, 5-b)$ So, $\frac{1+4+b}{3}=4 \Rightarrow b=7$ $\frac{2+a+5-b}{3}=1 \Rightarrow a=3$ $\therefore A=(1,2) ; B=(7,-2) ; C=(4,3)$ $\Delta=\frac{1}{2}\left|\begin{array}{ccc}1 & 1 & 2 \\ 1 & 7 & -2 \\ 1 & 4 & 3\end{array}\right|$ $\Delta=\frac{1}{2} \times 18=9$

In $\triangle \mathrm{ABC}$, if $\mathrm{A}=(1,2)$ and the equations of the medians through $B$ and $C$ are…

Solution