From a window h meters high above the ground in a street, the angles of elevation and depression of the top and foot of the other house on the opposite side of the street are α and β respectively. Show that the height of the opposite house is h(1+tanα cotβ) meters.
Answer:
- Let W be the window and AB be the house on the opposite side with height (h+h′) meters.
The figure below shows the given situation. - In the right-angled triangle AWP, we have cotα=WPAP⟹cotα=WPh′⟹WP=h′cotα…(i)
- In right-angled triangle WPB, we have cotβ=WPBP⟹cotβ=WPh⟹WP=hcotβ…(ii)
- On comparing eq (i) and eq (ii), we get h′cotα=hcotβ⟹h′=hcotβcotα=htanαcotβ
- Thus, height of the house = h+h′=h+htanαcotβ=h(1+tanαcotβ) meters.