In $\Delta ABC$,$m \angle A + m \angle B = m \angle C$. If $AC : BC = 3 : 4$ and $AB = 17.5$,find the perimeter of $\Delta ABC$.

In $\Delta ABC$,$m \angle A = 90^\circ$. If $b = 12$ and $c = 35$,then $a = \ldots$

The perimeter of rhombus $ABCD$ is $116$. If $AC = 42$, find $BD$.

In $\Delta ABC$,$D$ and $E$ are the midpoints of $\overline{BC}$ and $\overline{AC}$ respectively. $\overline{AD}$ and $\overline{BE}$ intersect at $G$. Line $m$ passing through $D$ and parallel to $\overline{BE}$ intersects $\overline{AC}$ at $K$. Then,$AC = \ldots$ (in $EK$)

The heights of two poles standing erect on a flat ground are $6 \, m$ and $11 \, m$. If the distance between their bases is $12 \, m$,find the distance between their tops.

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow ZXY$. If $AB = 12, BC = 8, CA = 10$ and $ZX = 10$,then $XY + YZ = \ldots$