Three points A , B , and C lie on a straight line on a horizontal plane such that B is the midpoint of AC . A vertical pole stands at a point O on the same plane. The angles of elevation of the top of the pole from A , B , and C are 30^ , 45^ , and 60^ respectively. If AC = 20 m, then the square of the height of the pole (in m ^2 ) is equal to

Step-by-step solution

Let the height of the vertical pole OP be h . The horizontal distances from the base of the pole O to the points A , B , and C can be found using the angles of elevation: OA = h 30^ = h 3 OB = h 45^ = h OC = h 60^ = h 3 In OAC , since B is the midpoint of AC , OB is the median to the side AC . We can apply Apollonius' Theorem: OA^2 + OC^2 = 2 (OB^2 + ( AC 2 )^2 ) Substitute the values into the theorem: (h 3 )^2 + ( h 3 )^2 = 2 (h^2 + 10^2 ) 3h^2 + h^2 3 = 2(h^2 + 100) 10h^2 3 = 2h^2 + 200 10h^2 3 - 2h^2 = 200 4h^2