Question

From an aeroplane, vertically above a straight horizontal road, the angle of depression of 2 consecutive milestones on opposite sides of the aeroplane are observed to be α and β . show that the height in miles of the aeroplane above the road is given by tan α tan β / tan α + tan β

Answer 1

Name the figure ABCD with AB as vertical height and BD as base with C in between. Now, AB/BC=tan $\alpha$ AB/BD= tan$\beta$ BCtan$\alpha$=BDtan$\beta$ BC tan$\alpha$=(BC+CD)tan $\beta$ BC (tan $\alpha$-tan$\beta$)= tan $\beta$ BC=tan $\beta$/(tan $\alpha$-tan $\beta$) AB=BC tan $\alpha$ Therefore AB=(tan $\alpha$ tan $\beta$)/(tan $\alpha$-tan $\beta$) Hence proved