The perimeter of square $ABCD$ is $48$. Find the length of its diagonal $\overline{AC}$. (in $\sqrt{2}$)

In $\Delta PQR$,$m \angle Q = 90^\circ$. If $PQ : QR = 1 : 1$,then $PQ : PR = \ldots$

In a quadrilateral $ABCD$,$\angle A + \angle D = 90^{\circ}$. Prove that $AC^{2} + BD^{2} = AD^{2} + BC^{2}$.

In $\Delta ABC$,$AB = 4$,$BC = 2\sqrt{3}$,and $AC = 2\sqrt{7}$. Then,in $\Delta ABC$,the length of the median on the longest side is $\ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $BM = 10$ and $CM = 5$,find the perimeter of $\Delta ABC$.

In a triangle,if the bisector of an angle bisects the side opposite to that angle,then prove that the triangle is an isosceles triangle.