In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 338$ and $AD = 5$,find $BC$.

$A$ $8 \, m$ long bamboo tree standing erect on the ground breaks at the height of $3 \, m$ from the ground. The broken part of the tree remains attached to the trunk. Find the distance in $m$ between the top of the tree on the ground and the base of the tree.

In $\square ABCD$,$P \in \overline{AB}$,$Q \in \overline{BC}$,$R \in \overline{CD}$,and $S \in \overline{AD}$ such that $\frac{AP}{AB} = \frac{CQ}{BC} = \frac{CR}{CD} = \frac{AS}{AD} = \frac{1}{3}$. Prove that $\square PQRS$ is a parallelogram.

In $\Delta XYZ$,$m\angle Y = 90^{\circ}$ and $\overline{YP}$ is a median. If $YP = 6$,then $XZ = \ldots$

In $\Delta PQR$,$m\angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude. If $PM = x$ and $RM = y$,find the lengths of $\overline{PQ}$,$\overline{QR}$,$\overline{PR}$,and $\overline{QM}$ in terms of $x$ and $y$.

Prove that the area of the semicircle drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the semicircles drawn on the other two sides of the triangle.