The area of the triangle formed by the lines $x + y - 3 = 0$,$x - 3y + 9 = 0$,and $3x - 2y + 1 = 0$ is:

If the coordinates of points $A, B, C$ are $(-1, 5), (0, 0)$ and $(2, 2)$ respectively,and $D$ is the midpoint of $BC$,then the equation of the perpendicular drawn from $B$ to the line $AD$ is:

Let the equations of two sides of a triangle be $3x - 2y + 6 = 0$ and $4x + 5y - 20 = 0$. If the orthocentre of this triangle is at $(1, 1)$,then the equation of its third side is

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 $\triangle ABC$?

$A$ regular octagon is formed by cutting congruent isosceles right-angled triangles from the corners of a square. If the square has side length $1$,the side length of the octagon is

The diagonals $AC$ and $BD$ of a rhombus $ABCD$ intersect at the point $O(3,4)$. If $BD=2 \sqrt{2}$, $A=(1,2)$, $B=(\alpha, \beta)$, $D=(\gamma, \delta)$ and $\alpha < \delta < \gamma < \beta$, then the value of $\beta+\gamma-\delta$ is: