Let $L_1$ be a line passing through $(2,1)$ and $(3, \frac{5}{2})$. $L_2$ is a line perpendicular to $L_1$ and passing through $(4,-1)$. The area of the triangle formed by $L_1$,$L_2$ and the $y$-axis is

Let the mirror image of point $A(\alpha, \beta)$ in the line mirror $x + 2y = 3$ be point $B$,and the image of $B$ in the line $3x - 2y = 5$ be $C$. If the origin is the orthocentre of triangle $ABC$ and $P(a, b)$ is a point inside the triangle such that triangles $PAB$,$PBC$,and $PCA$ have the same area,then $3(a + b)$ is:

$A$ vertex of a square is $(3, 4)$ and one diagonal lies on the line $x + 2y = 1$. Find the equation of the second diagonal which passes through the given vertex.

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

Area of the rhombus bounded by the four lines,$ax \pm by \pm c = 0$ is :

The perimeter of a triangle is $16 \text{ cm}$,and one of the sides is of length $6 \text{ cm}$. If the area of the triangle is $12 \text{ cm}^2$,then the triangle is: