In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. If $BD = 2 \sqrt{30}$ and $CD = 6$,then $AC = \ldots$

$A$ $8 \, m$ 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 a rectangle $ABCD$,if $AB^{2} + BC^{2} = 64$,then find the length of the diagonal $AC$.

In $\Delta ABC$,$\angle B$ is a right angle. Then,according to Pythagoras' theorem,$\ldots \ldots \ldots$ holds good.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 2x$,$BM = 3x + 5$,and $CM = 8x$,find the value of $x$.

In Figure $(i)$ and $(ii),$ $DE || BC.$ Find $EC$ in $(i)$ and $AD$ in $(ii).$