$A$ man having height $169 \; cm$ is standing near a pole. He casts a shadow $130 \; cm$ long. What is the length (in $cm$) of the pole if it casts a shadow $420 \; cm$ long?

The angle of elevation of the top of a tower at a point $G$ on the ground is $30^{\circ}$. On walking $20 \ m$ towards the tower,the angle of elevation becomes $60^{\circ}$. The height of the tower is equal to

$A$ man standing at a point $P$ (or $L$ in the diagram) is watching the top of a tower $N$,which makes an angle of elevation of $30^{\circ}$ with the man's eye. The man walks some distance towards the tower to point $Q$ to watch its top and the angle of elevation becomes $60^{\circ}$. What is the distance (in units) between the base of the tower $M$ and the point $P$ (or $L$)?

$A$ man is standing near a $6 \, m$ long pole which casts an $8 \, m$ long shadow. If the man casts a shadow of $2.4 \, m$,what is the height of the man (in $m$)?

If $x = a \cos \theta + b \sin \theta$ and $y = b \cos \theta - a \sin \theta,$ then $x^{2} + y^{2}$ is equal to

The tops of two poles of height $20 \ m$ and $14 \ m$ are connected by a wire. If the wire makes an angle $30^{\circ}$ with the horizontal,then the length of the wire is $m$.