$A$ farmer owns a triangular plot in Guntur. He measures the lengths of the sides of his property as $4 \text{ cm}$,$5 \text{ cm}$,and $7 \text{ cm}$. Then the area of land of the farmer in $\text{sq. cm}$ is

Consider a $\triangle ABC$ in the $XY$-plane with vertices $A=(0,0)$,$B=(1,1)$,and $C=(9,1)$. If the line $x=a$ divides the triangle into two parts of equal area,then $a$ equals

Let $A(\alpha, -2)$,$B(\alpha, 6)$,and $C\left(\frac{\alpha}{4}, -2\right)$ be the vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$,then which of the following is $NOT$ correct about $\triangle ABC$?

The area of the parallelogram formed by the lines $L_1 \equiv \lambda x+4 y+2=0$,$L_2 \equiv 3 x+4 y-3=0$,$L_3 \equiv 2 x+\mu y+6=0$,and $L_4 \equiv 2 x+y+3=0$,where $L_1$ is parallel to $L_2$ and $L_3$ is parallel to $L_4$,is

The area (in square units) of the triangle formed by the lines $x=0, y=0$ and $3x+4y=12$ is:

Suppose $ABOC$ is a rhombus in the first quadrant with $O$ being the origin. If the vertices $B$ and $C$ of $\triangle ABC$ lie respectively on $y=\frac{4}{3}x$ and $y=0$,and the side $BC$ passes through $\left(\frac{2}{3}, \frac{2}{3}\right)$,then the mid-point of $BC$ is