In $\Delta PQR$,the bisector of $\angle P$ intersects $\overline{QR}$ at $S$. If $QS = 2$,$PQ = 5$ and $SR = 3$,find $PR$.

$A$ girl of height $90\, cm$ is walking away from the base of a lamp-post at a speed of $1.2\, m/s$. If the lamp is $3.6\, m$ above the ground,find the length of her shadow after $4\, seconds$.

In the figure,$ABC$ is a triangle in which $\angle ABC < 90^{\circ}$ and $AD \perp BC$. Prove that $AC^{2} = AB^{2} + BC^{2} - 2BC \cdot BD$.

In $\Delta ABC$,$A-N-B$,$A-M-C$ and $B-X-C$. $\overline{XM} \parallel \overline{AB}$ and $\overline{XN} \parallel \overline{AC}$. $\overline{MN}$ intersects $\overline{CB}$ at $T$. Prove that $TX^2 = TB \times TC$.

$A$ guy wire attached to a vertical pole of height $18\, m$ is $24\, m$ long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

In $\Delta ABC$,$m\angle A + m\angle C = m\angle B$. If $AB = 7$ and $BC = 24$,then $AC = \ldots$