In square $ABCD$,$AB = 4 \sqrt{2}$. Then,the length of a diagonal of the square is......

State whether the following quadrilaterals are similar or not.

Two poles of heights $6 \, m$ and $11 \, m$ stand on a plane ground. If the distance between the feet of the poles is $12 \, m$,find the distance between their tops.

$ABC$ and $BDE$ are two equilateral triangles such that $D$ is the mid-point of $BC$. The ratio of the areas of triangles $ABC$ and $BDE$ is

Is the following statement true? Why?
"Two quadrilaterals are similar,if their corresponding angles are equal".

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.