Geometry

Trigonometry

Medium

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 9 m from the surface of the river, then find the width of the river.

$3(\sqrt{3} + 1)$

$5(\sqrt{3} + 1)$

$9(\sqrt{3} + 1)$

$12(\sqrt{3} + 1)$

✅ Correct Option: 3

Let the point on the bridge be $P$ at a height of $9$ m above the water surface.

Let the banks on opposite sides be $A$ and $B$ .

The angles of depression to banks $A$ and $B$ are $30°$ and $45°$ respectively.

By the property of alternate angles, the angles of elevation from the banks to point $P$ are also $30°$ and $45°$ .

Let $x$ be the horizontal distance from the point directly below $P$ to bank $A$ .

Using the tangent ratio:

$\tan(30°) = \dfrac{9}{x}$

$\dfrac{1}{\sqrt{3}} = \dfrac{9}{x}$

$x = 9\sqrt{3}$ m

Let $y$ be the horizontal distance from the point directly below $P$ to bank $B$ .

Using the tangent ratio:

$\tan(45°) = \dfrac{9}{y}$

$1 = \dfrac{9}{y}$

$y = 9$ m

The width of the river is the sum of both horizontal distances:

Width $= x + y$

Width $= 9\sqrt{3} + 9$

Width $= 9(\sqrt{3} + 1)$ m

Therefore, the width of the river is $9(\sqrt{3} + 1)$ m.

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