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.

The equation of the base of an equilateral triangle is $x+y=2$ and its opposite vertex is $(2,1)$. If $m_1, m_2$ are the slopes of the other two sides and the length of its side is $a$,then $|m_1-m_2|+a \sqrt{2}=$

If two vertices of an equilateral triangle are $(1, 0)$ and $(3, 0)$,and the third vertex lies in the first quadrant,find the area of the triangle.

Find the incenter of the triangle formed by the axes and the line $\frac{x}{a} + \frac{y}{b} = 1$.

The base of an isosceles triangle has its endpoints at $(2a, 0)$ and $(0, a)$. One side is parallel to the $y$-axis. Find the equation of the other side.

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: