The perimeter of equilateral $\Delta ABC$ is $24$. Find the length of its altitude. (in $\sqrt{3}$)

$A$ $4 \,m$ long ladder leans on a wall to reach a height of $3.2 \,m$ on the wall. Find the distance in $m$ between the lower end of the ladder and the base of the wall.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $\frac{AB}{5} = \frac{AC}{3}$ and $BD = 4.5$,find $BC$.

If $\Delta ABC \sim \Delta XYZ$ under the correspondence $ABC \leftrightarrow XZY$,and the perimeter of $\Delta ABC$ is $45$,the perimeter of $\Delta XYZ$ is $30$,and $AB = 21$,find the length of $XZ$.

$D$ and $E$ are points on the sides $AB$ and $AC$ of a triangle $ABC$ respectively,such that $AD = 2 \, cm$,$BD = 3 \, cm$,$BC = 7.5 \, cm$,and $DE \parallel BC$. Find the length of $DE$ (in $cm$).

In a triangle,if the bisector of an angle bisects the side opposite to that angle,then prove that the triangle is an isosceles triangle.