In $\Delta PQR, m\angle Q = 90^{\circ}$ and $\overline{QM}$ is a median. If $PQ = 20$ and $QR = 21$,find $QM$.

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 $\Delta ABC$,$AB = 4$,$BC = 2\sqrt{3}$,and $AC = 2\sqrt{7}$. Then,in $\Delta ABC$,the length of the median on the longest side is $\ldots$

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

Legs (sides other than the hypotenuse) of a right triangle are of lengths $16 \,cm$ and $8 \,cm$. Find the length of the side of the largest square that can be inscribed in the triangle. (in $cm$)

In $\Delta ABC$,$m \angle A = 90^{\circ}$. If $a = 20$ and $b = 12$,then $c = \dots$