$A$ $15 \, m$ long bamboo leans on a wall to reach the height of $12 \, m$ on the wall. Then,the lower end of the bamboo is $\ldots \ldots \, m$ away from the base of the wall.

$ABCD$ is a trapezium with $AB \parallel DC.$ $E$ and $F$ are points on non-parallel sides $AD$ and $BC$ respectively such that $EF \parallel AB$ (see Figure). Show that $\frac{AE}{ED} = \frac{BF}{FC}$.

$ABCD$ is a trapezium in which $AB \parallel DC$ and its diagonals intersect each other at the point $O$. Show that $\frac{AO}{BO} = \frac{CO}{DO}$.

In a square,the length of a diagonal is $8$. Find the length of the sides of the square. (in $\sqrt{2}$)

In the given figure,$\angle ADE = \angle AED$ and $\frac{AD}{DB} = \frac{AE}{EC}$. Prove that $\triangle ABC$ is an isosceles triangle.

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is a median. If $AC = 20$,then $BM = \ldots$