In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is a median. If $AB = 15$ and $BC = 20$,find $BM$.

In quadrilateral $ABCD$,$P$ is the midpoint of $\overline{BC}$. If $\overline{DP}$ produced intersects $\overline{AB}$ produced at $Q$,prove that $AB = 2CD$ given that $ABCD$ is a parallelogram.

In $\Delta ABC$ and $\Delta PQR$,$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$. If $AB = 15$,$PQ = 20$ and $BC = 12$,find $QR$.

In $\Delta ABC$,$m \angle A = 90^{\circ}$. If $AB = 3x - 2$,$AC = 5x + 4$,and $BC = 6x + 2$,find the value of $x$.

$\Delta ABC$ and $\Delta PQR$ are equilateral triangles. If $\frac{AB}{PQ} = \frac{3}{2}$,prove that $4 \times \text{Area of } \Delta ABC = 9 \times \text{Area of } \Delta PQR$.

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.