Question

In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively. Then the coordinates of $B$ and $C$ are (7,-2) , (4,3) respectively.If true enter 1 else o.

Answer 1

Mark the point $A(1,2)$ and trace the median $x+y=5$ throguh $B$. $x=4$ is the median through $C$ and hence the coordinates of $C$ be taken as $(4,a)$. Then the mid-point of $AC$ ie $(\frac{5}{2},\frac{a+2}{2})$ will lie on median through $B$ i.e., on $x+y=5$
$a+7=10$ $a=3$. Hence $C(4,3)$
Now choose the point $B$ to be $(h,k)$ which lies on $x+y=5$. $\therefore$ $h+k=5$. Also the mid point of $AB$ i.e., $(\frac{h+1}{2},\frac{k+2}{2})$ will lie on median through $C$ i.e, $x=4$
$\therefore \frac{h+1}{2}=4\Rightarrow h=7$ hence $k=-2$