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 $x$.

Hypotenuse of a right triangle is $25 \, cm$ and out of the remaining two sides,one is longer than the other by $5 \, cm$. Find the lengths of the other two sides.

If $\Delta ABC \sim \Delta PQR$ under the correspondence $ABC \leftrightarrow RPQ$,and $m\angle A = 35^{\circ}$ and $m\angle B = 50^{\circ}$,find $m\angle Q$. (in $^{\circ}$)

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.

In parallelogram $ABCD$,$AB^{2} + AD^{2} = 200$ and $BD = 12$. Then $AC = \ldots$

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $MA = 6$,$MB = 9$ and $MC = 8$,then $MD = \dots$