Two equal sides of an isosceles triangle are given by $7x-y+3=0$ and $x+y-3=0$. If the slope $m$ of the third side is an integer,then $m=$

Let $ABC$ be an isosceles triangle in which $A$ is at $(-1, 0)$,$\angle A = \frac{2\pi}{3}$,$AB = AC$ and $B$ is on the positive $x$-axis. If $BC = 4\sqrt{3}$ and the line $BC$ intersects the line $y = x + 3$ at $(\alpha, \beta)$,then $\frac{\beta^4}{\alpha^2}$ is :

Let the area of the triangle with vertices $A(1, \alpha)$,$B(\alpha, 0)$,and $C(0, \alpha)$ be $4 \text{ sq. units}$. If the points $(\alpha, -\alpha)$,$(-\alpha, \alpha)$,and $(\alpha^2, \beta)$ are collinear,then $\beta$ is equal to:

Which one of the following statements is true?

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

$A$ 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