Question

The angle of elevation of a cloud from a point h m above the level of water in a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$.Then the height of the cloud above the water level is $\frac{hsin(\beta -\alpha )}{sin(\beta +\alpha )}$

• A. True
• B. False

Answer 1

The correct option is A True

From Figure

Point $A$ is the cloud

Point $D$ is view point

$FD$ is parallel to $BE$ and $CD$ transversal

So $\angle FDC=\angle DCE$

From Reflection property

$\angle BCA=\angle DCE=\beta$

From $\Delta ABC$

$\sin \beta =\frac{AB}{AC}\cdots \left(1\right)$

From $\Delta CDE$

$\sin \beta =\frac{DE}{DC}\cdots \left(2\right)$

From $\left(1\right)\&\left(2\right)$

$⟹\frac{AB}{AC}=\frac{DE}{DC}$

$⟹AB=\frac{DE}{DC}\times AC\cdots \left(3\right)$

From $\Delta ACD$

By Exterior angle is sum of two interior opposite angles

$2\beta =\alpha +\beta +\angle DAC$

$\angle DAC=\beta -\alpha$

From $\sin$ rule

$⟹\frac{AC}{\sin (\alpha +\beta )}=\frac{DC}{\sin (\beta -\alpha )}$

$⟹\frac{AC}{DC}=\frac{\sin (\beta -\alpha )}{\sin (\alpha +\beta )}\cdots \left(4\right)$

Put it in $\left(3\right)$

$⟹AC=\frac{DE\sin (\beta -\alpha )}{\sin (\alpha +\beta )}$

From Question $DE=h$

$⟹AC=\frac{h\sin (\beta -\alpha )}{\sin (\alpha +\beta )}$