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 $x$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $AB = 11$ and $BC = 60$,find the perimeter of $\Delta ABC$.

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow DEF$. If $m \angle X = 50^{\circ}$ and $m \angle Y = 75^{\circ}$,then $m \angle F = \dots$ (in $^{\circ}$)

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AB = 3$ and $AC = 5$,find the length of $BM$.

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.

$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. The perimeter of $\Delta ABC$ is $35$ and the perimeter of $\Delta PQR$ is $28$. If $PR = 4\sqrt{10}$,then $AC = \ldots$