In $\triangle PQR$,$\angle R = \angle P$,$QR = 4 \, cm$,and $PR = 5 \, cm$. Then the length of $PQ$ is (in $cm$):

In $\Delta PQR$,$PM$ is a median. If $\text{ar}(\Delta PMQ) = 36 \text{ cm}^2$,then find $\text{ar}(\Delta PMR)$ in $\text{cm}^2$.

In $\Delta ABC$,$AB = AC$,$\angle A = 2x^{\circ}$ and $\angle B = x + 10^{\circ}$. Find each of the angles of $\Delta ABC$.

In a huge park,people are concentrated at three points (see Fig) :
$A$ : where there are different slides and swings for children,
$B$ : near which a man-made lake is situated,
$C$ : which is near to a large parking and exit. Where should an ice cream parlour be set up so that the maximum number of persons can approach it?
(Hint: The parlour should be equidistant from $A$,$B$,and $C$.)

Angles of a triangle are in the ratio $2: 4: 3$. The smallest angle of the triangle is (in $^{\circ}$)

In $\Delta ABC$,$AD$ is a median. If $\operatorname{ar}(ADB) = 53 \, cm^2$,then find $\operatorname{ar}(ABC)$ in $cm^2$.