Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x + y = 11$,$x + 2y = 16$,and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:

The base $BC$ of a triangle $ABC$ is bisected at the point $(p, q)$ and the equations of the sides $AB$ and $AC$ are $px + qy = 1$ and $qx + py = 1$. The equation of the median through $A$ is:

In a triangle $ABC$,if $BC=5, CA=6, AB=7$,then the length of the median drawn from $B$ onto $AC$ is

If the sides of a triangle are given by the equations $x + y - 5 = 0$,$x - y + 1 = 0$,and $y - 1 = 0$,then what is its circumcenter?

Let $B$ and $C$ be two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on the line $y-2x=2$ such that $\triangle ABC$ is an equilateral triangle. Then,the area of the $\triangle ABC$ is

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