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

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. Then,$BC$ is the geometric mean of..........

In the figure,if $DE \parallel BC,$ find the ratio of $\operatorname{ar}(ADE)$ and $\operatorname{ar}(DECB).$

In the figure,$ABC$ is a triangle right-angled at $B$ and $BD \perp AC$. If $AD = 4 \, cm$ and $CD = 5 \, cm$,find $BD$ and $AB$.

The correspondence $ABC \leftrightarrow QPR$ between $\Delta ABC$ and $\Delta PQR$ is a similarity. If $m \angle A : m \angle B : m \angle C = 15 : 8 : 7$,find the measures of all the angles of $\Delta PQR$.

$\Delta ABC$ and $\Delta XYZ$ are similar under the correspondence $ABC \leftrightarrow XZY$. If $AB = 6$,$XY = 12$,$YZ = 6$,and $ZX = 9$,find the perimeter of $\Delta ABC$.