$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB + BC = 7$,$XY + YZ = 10.5$,and $AC = 5$,then $XZ = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is a median. If $AB = 5$ and $BC = 12$,find $BD$.

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AC = 25$ and $AM = 16$,then $BM = \dots$

When a ladder leans on a wall to reach a height of $8 \, m$, its lower end rests $6 \, m$ away from the base of the wall. If the same ladder leans on the wall to reach a height of $6 \, m$, find the distance of its lower end from the base of the wall. (in $m$)

In $\Delta ABC$,$m \angle A = 90^{\circ}$ and $m \angle B = 30^{\circ}$. Then,which of the following holds true?

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x+1$,$MB = 2x-2$,$AN = 2x+2$ and $NC = 3x+1$,find the value of $x$.