If vertices of a quadrilateral are $A(0,0), B(3,4), C(7,7)$ and $D(4,3)$,then the quadrilateral $ABCD$ is

The orthocentre of the triangle formed by the lines $4x - 7y + 10 = 0$,$x + y = 5$,and $7x + 4y = 15$ is

Let a triangle be bounded by the lines $L_{1}: 2x + 5y = 10$,$L_{2}: -4x + 3y = 12$,and the line $L_{3}$,which passes through the point $P(2, 3)$,intersects $L_{2}$ at $A$ and $L_{1}$ at $B$. If the point $P$ divides the line segment $AB$ internally in the ratio $1:3$,then the area of the triangle is equal to

For a right-angled triangle having the lengths of two sides as $2 \sqrt{2}$ and $5$,find the possible lengths of the third side.

Let $ABC$ be a triangle with $A(-3, 1)$ and $\angle ACB = \theta$,where $0 < \theta < \frac{\pi}{2}$. If the equation of the median through $B$ is $2x + y - 3 = 0$ and the equation of the angle bisector of $C$ is $7x - 4y - 1 = 0$,then $\tan \theta$ is equal to:

All the vertices of a rectangle are of the form $(a, b)$ with $a, b$ integers satisfying the equation $(a-8)^2-(b-7)^2=5$. Then,the perimeter of the rectangle is