Height And Distance

Concept: This problem involves angles of depression and basic trigonometry in right-angled triangles. The angle of depression from an observer to an object is the angle between the horizontal line from the observer and the line of sight to the object, when the object is below the horizontal line. Step 1: Draw a diagram Let T be the top of the tower and F be its foot. The height of the tower TF = . Let the two objects be O1 and O2, on either side of the foot of the tower, in line with the foot. Let the horizontal line from T be TX. Angle of depression of O1 is . Angle of depression of O2 is . Since TX is parallel to the ground : (alternate interior angles). (alternate interior angles). We have two right-angled triangles: (right-angled at F) and (right-angled at F). The distance between the two objects is . Step 2: Calculate using In right-angled : Angle at is . Side opposite to (height) is TF = . Side adjacent to (base) is . We can use . So, . Step 3: Calculate using In right-angled : Angle at is . Side opposite to (height) is TF = . Side adjacent to (base) is . We can use . So, . Step 4: Calculate the distance between the objects Distance . Substitute the expressions for and : . Factor out : . This matches option (1).