In $\Delta PQR,$ $\overline{PS}$ is a median. If $PQ = 12$ and $PS = QS = 18.5,$ find $PR$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $a = 9$ and $c = 12$,then $b = \dots$

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle,write the length of its hypotenuse.
$7 \, cm, 24 \, cm, 25 \, cm$

In $\Delta ABC$,$A-P-B$,$A-Q-C$ and $\overline{PQ} \parallel \overline{BC}$. If $AP = 2$,$AB = 6$ and $AC = 9$,find $AQ$.

$\square ABCD$ is a trapezium in which $\overline{AB} \parallel \overline{CD}$. $M$ is a point on $\overline{AD}$ and $N$ is a point on $\overline{BC}$ such that $AM \times NC = BN \times MD$. If $\overline{MN}$ intersects $\overline{AC}$ at $O$,and $\frac{AM}{MD} = \frac{2}{3}$,find the ratio $\frac{AO}{AC}$.

$P$ and $Q$ are the points on the sides $DE$ and $DF$ of a triangle $DEF$ such that $DP = 5 \, cm$,$DE = 15 \, cm$,$DQ = 6 \, cm$,and $QF = 18 \, cm$. Is $PQ \parallel EF$? Give reasons for your answer.