In the figure,altitudes $AD$ and $CE$ of $\Delta ABC$ intersect each other at the point $P$. Show that $\Delta AEP \sim \Delta ADB$.

Using Theorem $6.2$,prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.

In $\Delta ABC$,$AB = 6\sqrt{3} \text{ cm}$,$AC = 12 \text{ cm}$,and $BC = 6 \text{ cm}$. The angle $B$ is (in $^o$):

$E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\Delta PQR$. For the given case,state whether $EF || QR$:
$PE = 4 \, cm, QE = 4.5 \, cm, PF = 8 \, cm$ and $RF = 9 \, cm$.

In the figure,two chords $AB$ and $CD$ of a circle intersect each other at point $P$ (when produced) outside the circle. Prove that $\Delta PAC \sim \Delta PDB$.

Nazima is fly fishing in a stream. The tip of her fishing rod is $1.8 \, m$ above the surface of the water and the fly at the end of the string rests on the water $2.4 \, m$ away from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut,how much string does she have out? If she pulls in the string at the rate of $5 \, cm$ per second,what will be the horizontal distance of the fly from her after $12 \, seconds$?