$A$ ladder $10\, m$ long reaches a window $8\, m$ above the ground. Find the distance of the foot of the ladder from the base of the wall. (in $, m$)

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 $^o$):

Sides $AB$ and $AC$ and median $AD$ of a triangle $ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another triangle $PQR$. Show that $\Delta ABC \sim \Delta PQR$.

$A$ ladder is placed against a wall such that its foot is at a distance of $2.5\, m$ from the wall and its top reaches a window $6\, m$ above the ground. Find the length of the ladder in $m$.

In an equilateral triangle $ABC$,$D$ is a point on side $BC$ such that $BD = \frac{1}{3} BC$. Prove that $9 AD^{2} = 7 AB^{2}$.