Two straight roads O A and O B intersect at O . A tower is situated within the angle formed by them and subtends angles of 45 o and 30 o at the points A and B where the roads are nearest to it. If O A = 100 meters and O B = 50 meters, then the height of the tower is

Step-by-step solution

Let, P Q be the tower of height h . Let, P A , P B be perpendiculars from P upon O A and O B respectively. Then, ∠ P A Q = 45 o and ∠ P B Q = 30 o O A = 100 , O B = 50 P A h = cot ⁡ 45 o = 1 ∴ P A = h P B h = cot ⁡ 30 o = 3 ∴ P B = 3 h Now, O P 2 = P A 2 + O A 2 = P B 2 + O B 2 ⇒ h 2 + 100 2 = 3 h 2 + 50 2 ⇒ 2 h 2 = 100 2 - 50 2 ⇒ h = 25 6 meters