$A$ quadrilateral has distinct integer side lengths. If the second-largest side has length $10$,then the maximum possible length of the largest side is

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

In a triangle $ABC$,$AD$ and $BE$ are medians. If $AD=4$,$\angle DAB = \frac{\pi}{6}$ and $\angle ABE = \frac{\pi}{3}$,then the area of $\triangle ABC$ is

Two sides of a triangle have lengths $a \text{ cm}$ and $b \text{ cm}$. If the triangle is to have the maximum area,then the length of the median from the vertex containing the sides $a$ and $b$ is

The triangle with vertices $A(2, 4)$,$B(2, 6)$,and $C(2 + \sqrt{3}, 5)$ is a . . . .

The vertices of a triangle are $(2, 1)$,$(5, 2)$ and $(4, 4)$. The lengths of the perpendiculars from these vertices to the opposite sides are