In $\Delta XYZ$,the bisector of $\angle X$ intersects $\overline{YZ}$ at $M$. If $XY = 6$,$XZ = 8$ and $YM = 4.2$,find $YZ$.

The perimeter of square $ABCD$ is $20$. Find $AC$. (in $\sqrt{2}$)

In quadrilateral $ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{O\}$. If $OA = 3x - 19$,$OB = x - 4$,$OC = x - 3$,and $OD = 4$,find the value of $x$.

As shown in the adjoining diagram of a staircase,the width of each step is $20 \, cm$ and the height is $15 \, cm$. Find the distance between points $A$ and $B$ in meters.

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow EDF$. If $XY = 3, YZ = 4, ZX = 6$ and $DF = 12$,find the perimeter of $\Delta DEF$.

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