In $\Delta PQR$,$\overline{PS}$ is a median. If $PQ = 9$,$PR = 40$ and $PS = 20.5$,find $QR$.

Out of the following triplets,$\ldots \ldots \ldots \ldots$ is not a Pythagorean triplet.

$\square ABCD$ 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 $\frac{AO}{AC}$.

In rectangle $ABCD$,$AC = 29$ and $AB + BC = 41$. Find $AB$ (given $AB > BC$).

$A$ $15\, m$ high tower casts a shadow $24\, m$ long at a certain time. At the same time,a telephone pole casts a shadow $16\, m$ long. Find the height of the telephone pole (in $m$).

In $\Delta ABC$,$D$,$E$ and $F$ are the midpoints of $\overline{BC}$,$\overline{CA}$ and $\overline{AB}$ respectively. Then,$\text{Area}(BDEF) = \ldots \times \text{Area}(\Delta ABC)$.