In a right triangle,if the measure of one of the angles is $45^{\circ}$,then the measure of the side opposite to the angle with measure $45^{\circ}$ is $\ldots \ldots \ldots$ times the measure of the hypotenuse.

As observed from the top of a $510 \ m$ high tower,the angles of depression of two houses situated in the east and west of the tower are found to be $30^{\circ}$ and $60^{\circ}$ respectively. Find the distance between the two houses (in $m$).

Observing an object from the point of observation,if one gets the angle of depression,then the object under observation is $\ldots \ldots \ldots . . .$

The angle of elevation of the top of a tower from a certain point is $30^{\circ}$. If the observer moves $20 \, m$ towards the tower,the angle of elevation of the top increases by $15^{\circ}$. Find the height of the tower.

$A$ flag-staff stands on the top of a building. The angle of elevation of the top of the building is $45^{\circ}$ and that of the top of the flag-staff is $60^{\circ}$,as observed by a man standing $15\, m$ away from the building. Find the height of the flag-staff from the top of the building (in $m$).

An observer $1.5\, m$ tall is $28.5\, m$ away from a tower. The angle of elevation of the top of the tower from his eyes is $45^\circ$. Then the height of the tower is ............ $m$.