In $\triangle ABC$,$m \angle B = 90^\circ$,$AB = 4$ and $BC = 3$. Find the radius of the incircle of the triangle.

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 122$ and $AD = 6$,find $BC$.

In $\Delta ABC$,$P, Q$ and $R$ are the midpoints of $\overline{BC}, \overline{CA}$ and $\overline{AB}$ respectively. Then,$Area(BCQR) = \ldots \ldots \ldots \times Area(ABC)$.

In the figure,$AD$ is a median of a triangle $ABC$ and $AM \perp BC$. Prove that:
$AB^{2} = AD^{2} - BC \cdot DM + \left(\frac{BC}{2}\right)^{2}$

In $\Delta ABC,$ $\overline{AD}$ is a median. If $AD=7,$ $AC=7$ and $BD=4,$ find the perimeter of $\Delta ABC.$

As shown in the adjoining diagram,$\angle B$ is an obtuse angle and $\overline{AM}$ is an altitude in $\Delta ABC$. Prove that $AC^{2} = AB^{2} + BC^{2} + 2 \cdot BC \cdot BM$.