Question

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at a height of 2.5 m from the banks, find width of the river.

Answer 1

Let $A$ and $B$ be two points on the banks on the opposite side of the river and $P$ be the point on the bridge at a height of 2.5 m.
Thus, we have:
$DP=2.5$ m, ∠PAD$={30}^o$ and ∠$PBD={45}^o$
In the right ∆$APD$, we have:
$\frac{DP}{AD}=\tan {30}^o=\frac{1}{\sqrt{3}}$
⇒ $\frac{2.5}{AD}=\frac{1}{\sqrt{3}}$
⇒ $AD=2.5\sqrt{3}$ m
In the right ∆$PDB$, we have:
$\frac{DP}{BD}=\tan {45}^o=1$
⇒ $\frac{2.5}{BD}=1$
⇒ $BD=2.5$ m
​
∴ Width of the river = $AB=(AD+BD)=(2.5\sqrt{3}+2.5)=6.83$ m